Mathematical Physics, Analysis and Geometry - Volume 14

Math Phys Anal Geom (2011) 14:1–20DOI 10.1007/s11040-010-9085-8

Higher Spin Dirac Operators Between Spacesof Simplicial Monogenics in Two Vector Variables

F. Brackx · D. Eelbode · L. Van de Voorde

Received: 30 November 2009 / Accepted: 20 October 2010 / Published online: 11 November 2010© Springer Science+Business Media B.V. 2010

Abstract The higher spin Dirac operator Qk,l acting on functions taking valuesin an irreducible representation space for so(m) with highest weight (k + 1

2 , l +12 , 1

2 , . . . , 12 ), with k, l ∈ N and k � l, is constructed. The structure of the kernel

space containing homogeneous polynomial solutions is then also studied.

Keywords Clifford analysis · Dirac operators · Higher spin

Mathematics Subject Classifications (2010) 15A66 · 30G35 · 22E46

1 Introduction

Consider an oriented spin manifold, i.e., a Riemannian manifold with a spinstructure which allows the construction of vector bundles whose underlyingsymmetry group is Spin(m) rather than SO(m), see e.g., [17]. On such aRiemannian spin manifold there is a whole system of conformally invariant,elliptic, first-order differential operators acting on sections of an appropriatespin bundle, whose existence and construction can be established through

F. Brackx · L. Van de Voorde (B)Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering,Ghent University, Ghent, Belgiume-mail: [email protected]

F. Brackxe-mail: [email protected]

D. EelbodeDepartment of Mathematics and Computer Science, University of Antwerp,Antwerp, Belgiume-mail: [email protected]

2 F. Brackx et al.

geometrical and representation theoretical methods, see e.g., [5, 13, 21, 22].In Clifford analysis these operators are studied from a function theoreticalpoint of view, mainly focusing on their rotational invariance with respect tothe spin group Spin(m), or its Lie algebra so(m), and considering functions onR

m instead of sections. The simplest example is the Dirac operator acting onspinor-valued functions; we refer to the standard references [1, 11, 15]. Next inline are the Rarita–Schwinger operator, acting on functions with values in theirreducible so(m)-representation with highest weight ( 3

2 , 12 , . . . , 1

2 ), and its gen-eralizations to the case of functions taking values in irreducible representationspaces with highest weight (k + 1

2 , 12 , . . . , 1

2 ), see e.g., [7, 8]. Also higher spinDirac operators acting on spinor-valued forms have been studied in detail, seee.g., [6, 20].

Our aim is to combine techniques from Clifford analysis and from repre-sentation theory, in order to investigate, from the function theoretical pointof view, general higher spin Dirac operators acting between functions takingvalues in an arbitrary finite-dimensional half-integer highest-weight represen-tation. As the case of the Rarita–Schwinger operator (and its generalizations)does not yet contain the seed from which the most general case can bederived, we study, in this paper, the particular case of the operator acting onfunctions taking values in the irreducible representation with highest weight(k + 1

2 , l + 12 , 1

2 , . . . , 12 ), with k, l ∈ N and k � l. This is done using the elegant

framework of Clifford analysis in several vector variables.

2 Clifford Analysis and Definitions

Let (e1, . . . , em) be an orthonormal basis for the Euclidean space Rm. We

denote by Cm the complex universal Clifford algebra, generated by thesebasis elements, its multiplication being governed by the relations e ie j +e je i = −2δij, i, j = 1, . . . , m. The space R

m is embedded in Cm by identifying(x1, . . . , xm) with the real Clifford vector x = ∑m

j=1 e jx j. The multiplication oftwo vectors x and y is given by xy = −〈x, y〉 + x ∧ y with

〈x, y〉 =m∑

j=1

x jy j and x ∧ y =∑

1�i< j�m

e ie j(xi y j − x jyi)

the scalar-valued Euclidean inner product and the bivector-valued wedgeproduct respectively. The wedge product of a finite number of vectors in R

m

may also be defined using the Clifford product:

Definition 1 The wedge product of N Clifford vectors x1,. . ., xN is defined as

x1 ∧ . . . ∧ xN := 1

N!∑

σ∈SN

sgn(σ ) x σ(1) . . . x σ(N),

where SN denotes the symmetric group in N elements.

Higher Spin Dirac Operators in Two Vector Variables 3

For convenience, we will work in odd dimension m = 2n + 1. In this casethere is a unique spinor space S, as opposed to the even-dimensional casem = 2n where there are two spinor representations (often referred to aseven and odd spinors). However, these cases do not differ from each otherconceptually: in case of even dimension m = 2n, it suffices to take intoaccount that the vector-valued (higher spin) Dirac operator will change theparity of the underlying values. The spinor space S should be thought ofas a minimal left ideal in Cm, which can be defined in terms of a primitiveidempotent; it is characterized by the highest weight ( 1

2 , . . . , 12 ) under the

standard multiplicative action of the spin group

Spin(m) =⎧⎨

⎩

2k∏

j=1

s j : k ∈ N , s j ∈ Sm−1

⎫⎬

⎭,

with Sm−1 the unit sphere in Rm. In case one prefers working with its Lie

algebra so(m), which can be identified with the subspace of bivectors in thealgebra Cm, the derived action should be used.

The Dirac operator is denoted ∂x = ∑mj=1 e j∂x j . It is an elliptic Spin(m)-

invariant first-order differential operator acting on spinor-valued functionsf (x) on R

m. It factorizes the Laplace operator: �x = −∂2x on R

m. An S-valuedfunction f is monogenic in an open region � ⊂ R

m if and only if it satisfies∂x f = 0 in �. For a detailed account of the theory of monogenic functions,so called Euclidean Clifford analysis, we refer the reader to e.g., [1, 11, 15].We also mention the Euler operator Ex = ∑

i xi∂xi , measuring the degree ofhomogeneity in the variable x.

Irreducible (finite-dimensional) modules for the spin group can be de-scribed in terms of spaces of traceless tensors satisfying certain symmetryconditions expressed in terms of Young diagrams, see e.g., [14, 16], but theycan also be realized in terms of vector spaces of polynomials, see e.g., [10, 15].We mention the following well-known examples from harmonic and Cliffordanalysis: the vector space Hk of C-valued harmonic homogeneous polynomialsof degree k ∈ N corresponds to the irreducible Spin(m)-module with highestweight (k, 0, . . . , 0), and the vector space Mk of spinor-valued monogenichomogeneous polynomials of degree k forms an irreducible representation ofSpin(m) with highest weight (k + 1

2 , 12 , . . . , 1

2 ).In what follows, N ∈ N and ∂ i is short for the Dirac operator ∂ui

.

Remark 1 In the sequel we will often need to refer to the highest weight ofa representation; to that end we introduce the short notation (λ1, . . . , λN)

for (λ1, . . . , λN, 0, . . . , 0) and denote by (λ1, . . . , λN)′ the highest weight (λ1 +12 , . . . , λN + 1

2 , 12 , . . . , 1

2 ).

Definition 2 A function f : RNm → C, (u1, . . . , uN) → f (u1, . . . , uN) is sim-

plicial harmonic if the following conditions are satisfied:

〈∂ i, ∂ j〉 f = 0, i, j = 1, . . . , N

〈ui, ∂ j〉 f = 0, 1 � i < j � N.

4 F. Brackx et al.

The vector space of C-valued simplicial harmonic polynomials, λi-homogeneous in the variable u i, will be denoted by Hλ1,...,λN (with λ1 � . . . �λN � 0 from now on).

Definition 3 A function f : RNm → S, (u1, . . . , uN) → f (u1, . . . , uN) is sim-

plicial monogenic if the following conditions are satisfied:

∂ i f = 0, i = 1, . . . , N

〈u i, ∂ j〉 f = 0, 1 � i < j � N.

The vector space of S-valued simplicial monogenic polynomials, λi-homogeneous in the variable ui, will be denoted by Sλ1,...,λN (with λ1 � . . . �λN � 0 from now on).

Remark 2 It is clear that if a function is simplicial monogenic in an open region� of R

Nm, then each of its scalar components is simplicial harmonic in �, or inother words: Sλ1,...,λN ⊂ Hλ1,...,λN ⊗ S.

Remark 3 The second condition in Definition 2 (respectively 3) implies thatan arbitrary polynomial pλ1,...,λN ∈ Hλ1,...,λN (respectively Sλ1,...,λN ) can be iden-tified with a C-valued (resp. S-valued) polynomial f depending only of anumber of specific wedge products of the vector variables:

pλ1,...,λN (u1, u2, . . . , uN) = f (u1, u1 ∧ u2, u1 ∧ u2 ∧ u3, . . . , u1 ∧ u2 ∧ . . . ∧ uN).

For details we refer to [10], where it is also shown that the space Hλ1,...,λN cor-responds to the irreducible Spin(m)-module with highest weight (λ1, . . . , λN),with respect to the regular representation H on C-valued simplicial harmonicpolynomials given by

H(s) f (u1, u1 ∧ u2, . . . , u1 ∧ . . . ∧ uN) = f (su1s, su1 ∧ u2s, . . . , su1 ∧ . . . ∧ uNs),

where s ∈ Spin(m). With respect to the regular representation L on S-valuedsimpicial monogenic polynomials, i.e.,

L(s) f (u1, u1 ∧ u2, . . . , u1 ∧ . . . ∧ uN) = sf (su1s, su1 ∧ u2s, . . . , su1 ∧ . . . ∧ uNs),

the space Sλ1,...,λN defines a model for the irreducible (finite-dimensional)Spin(m)-module with highest weight (λ1, . . . , λN)′.

Remark 4 As opposed to the one-variable case, the extra conditions inthe definition of simplicial monogenic polynomials, involving the operators〈ui, ∂ j〉, are needed in order to obtain an irreducible module for Spin(m). Forexample, Mk is an irreducible module, while Mλ1,λ2 := { f : R

2m → S | ∂1 f =∂2 f = 0} can be decomposed into irreducible modules, see e.g., [7], by meansof

Mλ1,λ2 =λ1−λ2⊕

j=0

〈u2, ∂1〉 jSλ1+ j,λ2− j.


From now on we take N = 2 and (u1, u2) = (u, v) in Definitions 2 and 3.Our object of interest is the elliptic, Spin(m)-invariant, first-order differentialoperator

Qk,l : C∞(Rm,Sk,l) → C∞(Rm,Sk,l)

f (x; u, v) → Qk,l f (x; u, v).

This higher spin Dirac operator Qk,l was already constructed in [12] followinga pragmatic approach. In this paper we will use techniques from representationtheory, which will ease the generalization to the most general case, anddescribe its polynomial solutions.

3 Refined Fischer Decomposition for Simplicial Monogenic Polynomials

We proceed as follows for the construction of the higher spin Dirac operatorQk,l. Let V be a representation of Spin(m) or its Lie algebra so(m). Denoteby �λ the finite-dimensional irreducible representation with highest weight λ.The multiplicity of �λ in V is denoted mλ(V) and the multiplicity of an arbitraryweight μ in �λ is denoted nμ(�λ). The following well-known result will be used(for the proof, we refer to e.g., [16]):

Proposition 1 If ν is a dominant integral weight such that mν(�λ ⊗ �μ) > 0,then there is a weight μ′ of �μ such that ν = λ + μ′ and mν(�λ ⊗ �μ) �nλ−ν(�μ).

One can then also prove the following theorem.

Theorem 1 For any pair of integers k � l � 0 with k > 0, one has

(k, l) ⊗ (0)′ = (k, l)′ ⊕ (1 − δl,0)(k, l − 1)′ ⊕ (1 − δk,l)(k − 1, l)′

⊕(1 − δl,0)(k − 1, l − 1)′.

Proof Take λ = (k, l), μ = (0)′ the highest weight for S and ν a dominantintegral weight such that mν(Hk,l ⊗ S) > 0. Then, by Proposition 1, there isa weight s of S such that ν = λ + s and mν(Hk,l ⊗ S) � ns(S) = 1. The possibleweights ν are given by

ν =(

k ± 1

2, l ± 1

2, ±1

2, . . . ,±1

2

)

.

As ν has to be a dominant integral weight, we only have to deal with thefollowing cases: ν = (k, l)′, ν = (k − 1, l)′, if k > l, and ν = (k, l − 1)′, ν =(k − 1, l − 1)′, if k � l > 0. The representations corresponding to these highestweights appear exactly once in Hk,l ⊗ S. We show this explicitly for ν = (k, l)′(the other cases being treated similarly). Let δ = (n − 1

2 , n − 32 , . . . , 3

2 , 12 ) be

6 F. Brackx et al.

half the sum of the positive roots and W the Weyl group. Using Klimyk’sformula, see e.g., [14], we find

mν(Hk,l ⊗ S) =∑

w∈Wsgn(w)nν+δ−w(λ+δ)(S) = nν+δ−1(λ+δ)(S) = n( 1

2 ,..., 12 )(S) = 1.

This follows from the fact that w = 1 ∈ W is the only element leading to a non-trivial result in the summation. Indeed, the action of W changes the sign of thecomponents λi of the weight (λ1, . . . , λn). In order to satisfy λ1 � . . . � λn, onlythe trivial action remains. This proves the claim. �

In case l = 0, the previous result encodes the Fischer decomposition forspinor-valued harmonic polynomials: Hk ⊗ S = Mk ⊕ uMk−1. This result iswell-known in Clifford analysis and states that any S-valued harmonic homo-geneous polynomial Hk of degree k in the vector variable u can be decomposedin terms of two monogenic homogeneous polynomials

Hk = Mk + u Mk−1,

with Mλ ∈ Mλ. The factor u in this formula is called an embedding factor: itrealizes an isomorphic copy of the irreducible module Mk−1 inside the tensorproduct Hk ⊗ S. Moreover, one can show that these polynomials are explicitlygiven by

Mk−1 = − 1

m + 2k − 2∂u Hk

Mk =(

1 + u ∂u

m + 2k − 2

)

Hk.

We will now generalize this result to the present setting k � l � 0. Theorem 1tells us how the space of S-valued simplicial harmonic polynomials Hk,l ⊗ S

decomposes into irreducible summands. It implies the existence of certainmaps which embed each of the spaces Sk,l, Sk−1,l, Sk,l−1 and Sk−1,l−1 (forappropriate k and l) into the space Hk,l ⊗ S. To ensure that these embeddingmaps are indeed morphisms realizing an isomorphic copy of the spaces ofsimplicial monogenics inside the space Hk,l ⊗ S, we have to check, next to thehomogeneity conditions, whether the conditions in Definition 2 are satisfied.Clearly, Sk,l ↪→ Hk,l ⊗ S is the trivial embedding. Also, it is easily verified that

u : Sk−1,l ↪→ Hk,l ⊗ S.

In order to embed the space Sk,l−1 into the tensor product Hk,l ⊗ S, it seemsobvious to start from the basic invariant v, as we need an embedding map ofdegree (0, 1) in (u, v), but this approach fails since 〈u, ∂v〉

[vSk,l−1

] = uSk,l−1 �=0. In order to obtain the required embedding map, it suffices to project onto thekernel of the operator 〈u, ∂v〉, which can be done by fixing c1 in the followingexpression:

v − c1 u〈v, ∂u〉 : Sk,l−1 ↪→ Hk,l ⊗ S.


For c1 = 1k−l+1 all conditions in Definition 2 are indeed satisfied. Similarly,

the last embedding map can be found as a suitable projection of a linearcombination of u v and v u, and is given by

〈v, u〉 − c2 v u − c3 〈u, u〉〈v, ∂u〉 : Sk−1,l−1 ↪→ Hk,l ⊗ S

with c2 = −m+k+l−4m+2k−4 and c3 = 1

m+2k−4 . This can be summarized as follows:

Proposition 2 For any pair of integers k � l � 0 with k > 0, one has

Hk,l ⊗ S = Sk,l ⊕ (1 − δl,0)νk,l Sk,l−1 ⊕ (1 − δk,l)μk,l Sk−1,l

⊕ (1 − δl,0)κk,l Sk−1,l−1,

with the embedding maps

νk,l := v − u〈v, ∂u〉k − l + 1

: Sk,l−1 ↪→ Hk,l ⊗ S

μk,l := u : Sk−1,l ↪→ Hk,l ⊗ S

κk,l := 〈v, u〉 + m + k + l − 4

m + 2k − 4v u − 〈u, u〉〈v, ∂u〉

m + 2k − 4: Sk−1,l−1 ↪→ Hk,l ⊗ S.

Remark 5 The embedding map μk,k clearly does not exist, in view of thedominant weight condition. If l = 0 the embedding maps νk,l and κk,l do notexist neither.

Let k > l > 0 and suppose ψ ∈ Hk,l ⊗ S. According to Proposition 2, thereexists ψp,q ∈ Sp,q such that

ψ = ψk,l + νk,l ψk,l−1 + μk,l ψk−1,l + κk,l ψk−1,l−1. (1)

An explicit expression for the projection operators on each of the summandsinside Hk,l ⊗ S can then be obtained as follows. First, the action of ∂v on (1)annihilates two summands and leads to

∂vψ = − (m + 2l − 4) ψk,l−1

+(

1 − m + k + l − 4

m + 2k − 4(m + 2l − 2)

)

u ψk−1,l−1. (2)

Acting again with ∂u, we find

ψk−1,l−1 = (m + 2k − 4) ∂u∂vψ

(m + 2k − 2)(m + 2l − 4)(m + k + l − 3).

This gives rise to a projection operator πk−1,l−1, defined as

πk−1,l−1 : Hk,l ⊗ S → Sk−1,l−1

ψ → (m + 2k − 4) ∂u∂vψ

(m + 2k − 2)(m + 2l − 4)(m + k + l − 3). (3)

8 F. Brackx et al.

Substituting the expression for ψk−1,l−1 in (2), we find

ψk,l−1 = − 1

m + 2l − 4

(

1 + u ∂u

m + 2k − 2

)

∂vψ,

which defines a second projection operator πk,l−1:

πk,l−1 : Hk,l ⊗ S → Sk,l−1

ψ → − 1

m + 2l − 4

(

1 + u ∂u

m + 2k − 2

)

∂vψ.

Finally, using the previous results, the action of ∂u on (1) leads to

ψk−1,l = − 1

m + 2k − 2

[(

1 +(

k − lk − l + 1

)v ∂v

m + 2l − 4

)

∂u

+ 1

k − l + 1

(

1 + u ∂u

m + 2l − 4

)

〈v, ∂u〉∂v

]

ψ,

which defines the third projection operator:

πk−1,l : Hk,l ⊗ S → Sk−1,l

ψ → − 1

m + 2k − 2

[(

1 +(

k − lk − l + 1

)v ∂v

m + 2l − 4

)

∂u

+ 1

k − l + 1

(

1 + u ∂u

m + 2l − 4

)

〈v, ∂u〉∂v

]

ψ.

The last projection operator on the summand Sk,l is then given by

πk,l := 1 − πk−1,l − πk,l−1 − πk−1,l−1.

We gather all this information in the following theorem.

Theorem 2 (Refined Fischer Decomposition for Simplicial Monogenics) EachS-valued simplicial harmonic polynomial Hk,l in two vector variables can beuniquely decomposed in terms of simplicial monogenic polynomials:

Hk,l = Mk,l + νk,l Mk,l−1 + μk,l Mk−1,l + κk,l Mk−1,l−1,

with the embedding maps def ined in Proposition 2 and with

Mk,l = πk,l(Hk,l) Mk,l−1 = πk,l−1(Hk,l)

Mk−1,l = πk−1,l(Hk,l) Mk−1,l−1 = πk−1,l−1(Hk,l).

Proof Only the uniqueness has to be addressed, but this can easily be proved. �

4 Construction of the Operator Qk,l

We now use the refined Fischer decomposition of Theorem 2 to construct thehigher spin Dirac operator Qk,l. Since the classical Dirac operator ∂x can be


seen as an endomorphism on S-valued functions, the action of ∂x on a Hk,l ⊗ S-valued function preserves the values. This gives rise to a collection of invariantoperators defined through the following diagram:

C∞(Rm,Hk,l ⊗ S)∂x

��C∞(Rm,Hk,l ⊗ S)

= =C∞(Rm,Sk,l)

Qk,l��

T k,lk,l−1

��

T k,lk−1,l

��

��

C∞(Rm,Sk,l)

⊕ ⊕C∞(Rm, νk,lSk,l−1) C∞(Rm, νk,lSk,l−1) .

⊕ ⊕C∞(Rm, μk,lSk−1,l) C∞(Rm, μk,lSk−1,l)

⊕ ⊕C∞(Rm, κk,lSk−1,l−1) C∞(Rm, κk,lSk−1,l−1)

The non-existence of an invariant operator from C∞(Rm,Sk,l) toC∞(Rm, κk,lSk−1,l−1) (i.e., the dotted arrow in the diagram above) can beproved by means of the construction method of conformally invariantoperators using generalized gradients, see e.g., [13, 22]. It essentially followsfrom the fact that the tensor product Sk,l ⊗ C

m does not contain the summandSk−1,l−1. The next lemma shows that this can also be verified through directcalculations in Clifford analysis.

Lemma 1 For every f ∈ C∞(Rm,Sk,l) one has ∂u∂v∂x f = 0.

Proof The definition of the Euclidean inner product leads to

∂u∂v∂x f = −2∂u〈∂v, ∂x〉 f − ∂u∂x∂v f = 0,

since ∂u f = ∂v f = 0. �

Hence, it follows from (3) that πk−1,l−1(∂x f ) ≡ 0, for every f ∈ C∞(Rm,Sk,l).An explicit expression for the operators Qk,l, T k,l

k,l−1 and T k,lk−1,l in the diagram

above is then obtained using results of the previous section.

Definition 4 For all integers k � l � 0 with k > 0, there are (up to a multiplica-tive constant) unique invariant first-order differential operators Qk,l definedby

Qk,l : C∞(Rm,Sk,l) → C∞(Rm,Sk,l) : f → πk,l(∂x f ),

10 F. Brackx et al.

or explicitly,

Qk,l f =[

1 + u ∂u

m + 2k − 2+ v ∂v

m + 2l − 4− 2

u〈v, ∂u〉∂v

(m + 2k − 2)(m + 2l − 4)

]

∂x f.

In case k = l > 0, the operators reduce to

Qk,k f =[

1 + (v − u〈v, ∂u〉)∂v

m + 2k − 4

]

∂x f.

Remark 6 In case l = 0, we find the Rarita–Schwinger operators Rk, as definedin [7]:

Qk,0 = Rk =(

1 + u ∂u

m + 2k − 2

)

∂x. (4)

The ellipticity of this operator Qk,l follows e.g., from [5], and the Spin(m)-invariance can be expressed through the following commutative diagram:

C∞(Rm,Sk,l)Qk,l

��

L(s)

��

C∞(Rm,Sk,l)

L(s)

��

��

C∞(Rm,Sk,l)Qk,l

�� C∞(Rm,Sk,l)

with L(s)(

f (u, v)) = sf (sus, svs) the natural action of Spin(m) on higher spin

fields.Similar calculations lead to the so-called dual twistor operators, which

are visualized as the diagonal arrows in the diagram above. We adopt theconvention that each operator of twistor-type will be denoted by means ofthe letter T , together with upper and lower indices. The upper indices denotethe highest weight of the source space, whereas the lower indices denote thehighest weight of the target space (discarding the half-integers).

Definition 5 For all integers k > l � 0, the dual twistor operators T k,lk−1,l are

defined as the unique invariant first-order differential operators

T k,lk−1,l : C∞(Rm,Sk,l) → C∞(Rm, μk,lSk−1,l) : f → μk,lπk−1,l(∂x f ),


or explicitly, hereby taking Lemma 1 into account:

T k,lk−1,l f = μk,l

[2

m + 2k − 2

(

〈∂u, ∂x〉 + 〈v, ∂u〉〈∂v, ∂x〉k − l + 1

)

f]

.

In case k = l > 0, these operators do not exist.

Definition 6 For all integers k > l > 0, the dual twistor operators T k,lk,l−1 are

defined as the unique invariant first-order differential operators

T k,lk,l−1 : C∞(Rm,Sk,l) → C∞(Rm, νk,lSk,l−1) : f → νk,lπk,l−1(∂x f ).

or explicitly, again taking Lemma 1 into account:

T k,lk,l−1 f = νk,l

(2

m + 2l − 4〈∂v, ∂x〉 f

)

.

In case k = l > 0, these operators reduce to

T k,kk,k−1 f = 2

m + 2k − 4

(v − u〈v, ∂u〉

)〈∂v, ∂x〉 f.

Note that these operators are called dual, because there exist also twistoroperators acting in the opposite direction; these ones are given below, but theywill not be explicitly used in this paper.

Remark 7 The twistor operators T k−1,lk,l and T k,l−1

k,l are defined as

T k−1,lk,l : C∞(Rm, μk,lSk−1,l) → C∞(Rm,Sk,l) : μk,l f → πk,l(∂xμk,l f )

and

T k,l−1k,l : C∞(Rm, νk,lSk,l−1) → C∞(Rm,Sk,l) : νk,l f → πk,l(∂xνk,l f ).

Remark 8 We introduce the following short notations for the dual twistoroperators without the embedding factor:

Tk,lk,l−1 := (νk,l)

−1T k,lk,l−1 = 〈∂v, ∂x〉

Tk,lk−1,l := (μk,l)

−1T k,lk−1,l = 〈∂u, ∂x〉 + 〈v, ∂u〉〈∂v, ∂x〉

k − l + 1.

5 Constructing Polynomial Null Solutions

As in any function theory linked to a differential operator, a crucial piece ofknowledge is the full description of its polynomial null solutions. This will bethe subject of this section. Note that in this respect, higher spin Dirac operatorsbehave completely different from the classical Dirac operator: the spaces ofpolynomial null solutions will no longer be irreducible as a Spin(m)-module,and a typical problem is to decompose polynomial kernel spaces for higherspin Dirac operators into irreducibles.

12 F. Brackx et al.

Let us denote by KerhD the vector space of h-homogeneous polynomialnull solutions of the (linear differential) operator D. As a function f belongsto KerhQk,l if and only if it satisfies πk,l(∂x f ) = 0, there are two possibilities tosatisfy this condition. This gives rise to two types of homogeneous polynomialnull solutions f for Qk,l: either ∂x f = 0 (called type A solutions) or ∂x f �= 0but πk,l(∂x f ) = 0 (called type B solutions). We will now treat each of thesepossibilities in detail.

5.1 Solutions of Type A

For all integers h � k � l > 0, define Ph,k,l(S) to be the space of S-valuedpolynomials in three vector variables (x, u, v), homogeneous of degree h, kand l in x, u and v respectively. Denote the subspace of triple monogenics by

Mh,k,l = { f ∈ Ph,k,l(S) | ∂x f = ∂u f = ∂v f = 0}.This vector space is highly reducible with respect to the action of Spin(m), andin [4] we have determined the decomposition of this space in terms of irre-ducible Spin(m)-modules, making use of the fact that each vector space Sp,q,r

can be seen as a highest weight vector for the algebra gl3, with positive rootvectors {〈x, ∂u〉, 〈x, ∂v〉, 〈u, ∂v〉}. The vector space Ms

h,k,l = Mh,k,l ∩ Ker〈u, ∂v〉,or more explicitly Ms

h,k,l = { f ∈ Mh,k,l | 〈u, ∂v〉 f = 0}, is, by construction, pre-cisely the space of h-homogeneous solutions for Qk,l of type A. The decom-position of this space into irreducible spaces for Spin(m) was also determinedin [4], using branching rules from gl3 to gl2. Using the so-called raising andlowering operators 〈u, ∂x〉 and 〈v, ∂x〉(Eu − Ev) − 〈u, ∂x〉〈v, ∂u〉, which werestudied in the much broader setting of transvector algebras and weight basesfor Lie algebras in e.g., [18], we were able to prove that for k � l,

Msh,k,l =

k−l⊕

i=0

l⊕

j=0

〈u, ∂x〉i [〈v, ∂x〉(Eu − Ev) − 〈u, ∂x〉〈v, ∂u〉] j Sh+i+ j,k−i,l− j. (5)

To lighten the notation, we will often omit these (commuting) embeddingfactors and denote the irreducible modules in these decompositions by theirhighest weights only:

Msh,k,l =

k−l⊕

i=0

l⊕

j=0

(h + i + j, k − i, l − j)′. (6)

In the special case where l = 0, we reobtain the results mentioned earlier inRemark 4:

Mh,k := Msh,k,0 =

k⊕

i=0

(h + i, k − i)′. (7)


Remark 9 Every irreducible module in Msh,k,l appears with multiplicity one,

which is a general fact for branching rules from gln to gln−1. A necessary condi-tion for a module Sp,q,r to be, up to an isomorphic copy, in the decompositionof Ms

h,k,l is p + q + r = h + k + l.

5.2 Solutions of Type B

A different approach is required to describe the type B solutions. It is in-structive to recall the Rarita–Schwinger case. Let f (x; u) be a polynomial inC∞(Rm,Mk), homogeneous of degree h � k in the vector variable x. In [7],the following equivalence was proved:

f ∈ KerhRk ⇔{

∂x f = ug∂u f = 0

,

with g ∈ Kerh−1Rk−1. In order for this inhomogeneous system of equations(involving the classical Dirac operator) to have a solution f , certain conditionson g must be satisfied. The necessary and sufficient conditions (called compat-ibility conditions for short) under which an inhomogeneous system in severalDirac operators has a solution, were thoroughly studied in [9]. Referring to[7, 9] for details, the compatibility conditions for the existence of a solutionfor the system above are �u(ug) = 0 and ∂u∂x(ug) = 0. The first condition isequivalent with the monogeneity of g in the variable u, i.e., g ∈ C∞(Rm,Mk−1),whereas the second compatibility condition is equivalent to g ∈ Kerh−1Rk−1.In other words: these compatibility conditions signify that the kernel spacefor the operator Rk−1 can be embedded into the kernel space for Rk, usingcertain inversion operators which were described in [7]. Type B solutions forRk are thus equivalent with elements of Kerh−1Rk−1; their structure may thusbe described through an inductive procedure.

In the case of the operator Qk,l, we have the equivalence

Qk,l f = 0 ⇔ ∂x f = μk,l2

m + 2k − 2T

k,lk−1,l f + νk,l

2

m + 2l − 4T

k,lk,l−1 f. (8)

Moreover, we also have the following results, which can be proved by directcalculations:

Proposition 3 For any couple of integers k > l > 0, resp. k � l > 0, andfor any function f ∈ C∞(Rm,Sk,l), one has: πk−1,l(∂xT

k,lk,l−1 f ) = 0, resp.

πk,l−1(∂xTk,l

k−1,l f ) = 0.

This means that non-trivial maps C∞(Rm, μk,lSk−1,l) → C∞(Rm, νk,lSk,l−1)

(and vice versa) do not exist. They are visualized by the dotted lines in the

14 F. Brackx et al.

diagram below, where the double action of the Dirac operator on C∞(Rm,Sk,l)

is considered.

C∞(Rm,Hk,l ⊗ S) •∂x

��•∂x

��• C∞(Rm,Hk,l ⊗ S)

= = =C∞(Rm,Sk,l) •

Qk,l��

T k,lk,l−1

��

T k,lk−1,l

��

��

��

��

�� • ��

��

��

��

��

��

�� • C∞(Rm,Sk,l)

⊕ ⊕ ⊕C∞(Rm, νk,lSk,l−1) • •

��

��

��

��

��

��

�� • C∞(Rm, νk,lSk,l−1)

⊕ ⊕ ⊕C∞(Rm, μk,lSk−1,l) • •

��

��

�� • C∞(Rm, μk,lSk−1,l)

⊕ ⊕ ⊕C∞(Rm, κk,lSk−1,l−1) • • • C∞(Rm, κk,lSk−1,l−1)

Proposition 4 Let f ∈ KerhQk,l .

(i) If k � l > 0, then Tk,lk,l−1 f ∈ Kerh−1Qk,l−1.

(ii) If k > l � 0, then Tk,lk−1,l ∈ Kerh−1Qk−1,l .

Proof Again, a straightforward calculation leads to the desired result. Let c1 =m + 2k − 2 and c2 = m + 2l − 4. For every f ∈ KerhQk,l, we have

〈∂v, ∂x〉Qk,l f = 0 ⇔ 〈∂v, ∂x〉[c1c2 + c2 u ∂u + c1 v ∂v − 2u〈v, ∂u〉∂v

]∂x f = 0

⇔ Qk,l−1〈∂v, ∂x〉 f = 0.

This may also be proved by considering the double action of the Diracoperator. Since ∂2

x is scalar, the following implication obviously holds:

f ∈ C∞(Rm,Sk,l) ⇒ ∂2x f ∈ C∞(Rm,Sk,l).

Therefore, the projection on each of the other summands in the decompositionof Hk,l ⊗ S is zero. In particular, we have that πk,l−1(∂

2x f ) = 0, which in

combination with Proposition 3 leads to the following identity (up to a suitableconstant):

T k,lk,l−1Qk,l + νk,lQk,l−1(νk,l)

−1T k,lk,l−1 = 0. (9)


This can be visualized by the parallelogram formed by the double arrows inthe diagram below:

C∞(Rm,Hk,l ⊗ S) •

∂2x

∂x��•

∂x��• C∞(Rm,Hk,l ⊗ S)

= = =C∞(Rm,Sk,l) •

Qk,l��

T k,lk,l−1

��

��

T k,lk−1,l

��

��

��

��

� •

��

��

��

��

��

��

� • C∞(Rm,Sk,l)

⊕ ⊕ ⊕C∞(Rm, νk,lSk,l−1) • • ��• C∞(Rm, νk,lSk,l−1)

⊕ ⊕ ⊕C∞(Rm, μk,lSk−1,l) • • �� • C∞(Rm, μk,lSk−1,l)

⊕ ⊕ ⊕C∞(Rm, κk,lSk−1,l−1) • • • C∞(Rm, κk,lSk−1,l−1)

For f ∈ KerhQk,l, the identity (9) reduces to νk,lQk,l−1Tk,lk,l−1 f = 0, prov-

ing the first statement. The calculations for proving the second statementare somewhat more technical and involved, but using operator identities inClifford analysis one can verify that

〈∂u, ∂x〉Qk,l f + 2

c1 − c2〈v, ∂u〉〈∂v, ∂x〉Qk,l f = 0

⇒ Qk−1,l

(

〈∂u, ∂x〉 + 〈v, ∂u〉〈∂v, ∂x〉k − l + 1

)

f = 0,

which leads to the desired statement. Invoking once more Proposition 3, thiscan also be proved by considering the parallelogram formed by the dashedlines in the diagram above, leading to the identity

T k,lk−1,lQk,l + μk,lQk−1,l(μk,l)

−1T k,lk−1,l = 0,

For f ∈ KerhQk,l, this leads to μk,lQk−1,lTk,lk−1,l f = 0, which concludes the

proof. �

In view of Proposition 4, the following implication holds:

f ∈ KerhQk,l ⇒ ∂x f = μk,lg1 + νk,lg2

with g1 ∈ Kerh−1Qk−1,l and g2 ∈ Kerh−1Qk,l−1. Conversely, let f ∈ C∞(Rm,Sk,l)

now be a polynomial, h-homogeneous in x, such that g1 and g2 satisfy therequirements mentioned above and with

∂x f = μk,lg1 + νk,lg2, (10)

16 F. Brackx et al.

then also f ∈ KerhQk,l. Now we would like to investigate whether for anychoice of these polynomials g1 and g2, there indeed exists a polynomial fsatisfying (10). In other words, we are trying to characterize the conditionswhich have to be imposed on g1 ∈ Kerh−1Qk−1,l and g2 ∈ Kerh−1Qk,l−1, suchthat the following equivalence holds:

f ∈ KerhQk,l ⇔

⎧⎪⎪⎨

⎪⎪⎩

∂x f = μk,lg1 + νk,lg2

∂u f = 0∂v f = 0〈u, ∂v〉 f = 0

.

Just like for the Rarita–Schwinger case, see [7], this requires the study ofcompatibility conditions for an inhomogeneous system of equations involvingthree Dirac operators. The system above is not of the form considered in[9] due to the presence of the last equation. We will split this system intoa simplified system and an extended system. The simplified system (denotedSiSy) is given by

SiSy ↔⎧⎨

⎩

∂x f = μk,lg1 + νk,lg2

∂u f = 0∂v f = 0

,

whereas adding the extra condition 〈u, ∂v〉 f = 0 defines the extended system(denoted ExSy). The next proposition tells us it is sufficient to study solutionsfor SiSy:

Proposition 5 Let f ∈ Ph,k,l(S) be a solution of SiSy. Then the projection π( f )of f on the kernel of 〈u, ∂v〉 satif ies ExSy:

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

∂xπ( f ) = μk,lg1 + νk,lg2

∂uπ( f ) = 0

∂vπ( f ) = 0

〈u, ∂v〉π( f ) = 0

.

Proof Using the Fischer decomposition with respect to the operator 〈u, ∂v〉,we can write any solution f of SiSy as

f = fk,l + 〈v, ∂u〉 fk+1,l−1 + . . . + 〈v, ∂u〉l fk+l,0 =l∑

j=0

〈v, ∂u〉 j fk+ j,l− j

with fk+ j,l− j in Ker〈u, ∂v〉 ⊂ Ph,k,l(S), for all j = 0, . . . , l. Define the projectionmap π by

π : Ph,k,l(S) → Ker〈u, ∂v〉 : f → π( f ) = fk,l.

We will prove that fk,l satisfies ExSy. Because ∂u f = 0 and [∂u, 〈v, ∂u〉] =0, we already have that ∂u fk+ j,l− j = 0. Combining this result with ∂v f = 0,which means that also the commutator [∂v, 〈v, ∂u〉] = ∂u acts trivially, we find


that ∂v fk+ j,l− j = 0 holds too. Finally, we verify that ∂x fk,l = μk,lg1 + νk,lg2. As[∂x, 〈u, ∂v〉] = 0, it is easily seen that π(∂x f ) = ∂xπ( f ) and hence

∂x fk,l = ∂xπ( f ) = π(∂x f ) = π(μk,lg1 + νk,lg2) = μk,lg1 + νk,lg2

because μk,lg1 + νk,lg2 ∈ Ker〈u, ∂v〉 by construction. Note that fk,l �= 0, sinceotherwise μk,lg1 + νk,lg2 = 0. �

To any inhomogeneous system of Dirac equations of the form⎧⎪⎨

⎪⎩

∂x f = h1

∂u f = h2

∂v f = h3

corresponds the following set of compatibility conditions, see [9]:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

�uh1 + ∂x∂uh2 = 0

�vh1 + ∂x∂vh3 = 0

�vh2 + ∂u∂vh3 = 0

�xh2 + ∂u∂xh1 = 0

�xh3 + ∂v∂xh1 = 0

�uh3 + ∂v∂uh2 = 0

{∂x, ∂u}h3 = ∂v(∂xh2 + ∂uh1)

{∂u, ∂v}h1 = ∂x(∂uh3 + ∂vh2)

{∂v, ∂x}h2 = ∂u(∂vh1 + ∂xh3)

.

The last three relations (which are linear dependent) are the radial algebrarelations, which were investigated in [19]. In our present case of interest, wehave to put h1 = μk,lg1 + νk,lg2 and h2 = 0 = h3. Motivated by the Rarita–Schwinger case, we will split these conditions into two sets. First of all, definethe compatibility conditions of type I (denoted CC-I):

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

�u(μk,lg1 + νk,lg2) = 0

�v(μk,lg1 + νk,lg2) = 0

∂u∂v(μk,lg1 + νk,lg2) = 0

∂v∂u(μk,lg1 + νk,lg2) = 0

{∂u, ∂v}(μk,lg1 + νk,lg2) = 0

together with the extra condition (which is the one leading to the ExSy)

〈u, ∂v〉(μk,lg1 + νk,lg2) = 0.

We are then left with two compatibility conditions of type II (denoted CC-II):{

∂u∂x(μk,lg1 + νk,lg2) = 0 (i)

∂v∂x(μk,lg1 + νk,lg2) = 0 (ii).

18 F. Brackx et al.

Note that they are not independent, because

∂u∂x(μk,lg1 + νk,lg2) = 0 ⇒ 〈u, ∂v〉∂u∂x(μk,lg1 + νk,lg2) = 0

⇔ ∂v∂x(μk,lg1 + νk,lg2) = 0.

This means that it is sufficient to check CC-II (i). However, it will turn out tobe useful to investigate both conditions anyway.

We mentioned before that in the case of the Rarita–Schwinger operator,the analysis of compatibility conditions leads to the conclusion that the kernelspace for Rk−1 can be embedded into the kernel space for Rk. The compatibil-ity conditions exactly determine the structure of the kernel space (i.e., the typeB solutions).

In the present case of the operator Qk,l, it is not difficult to show thatthe conditions of CC-I are equivalent with g1 and g2 being elements ofC∞(Rm,Sk−1,l) and C∞(Rm,Sk,l−1) respectively. In other words, these condi-tions again fix the values. However, the conditions of CC-II are not equivalentwith every g1 ∈ Kerh−1Qk−1,l and every g2 ∈ Kerh−1Qk,l−1. Indeed, we willprove that only for g1 and g2 satisfying

Tk−1,lk−1,l−1g1 = (m + 2l − 4)(k − l + 2)

(m + 2k − 2)(k − l + 1)T

k,l−1k−1,l−1g2 (11)

there exists a polynomial f in KerhQk,l such that ∂x f = μk,lg1 + νk,lg2. Inparticular, this relation is satisfied for g1 and g2 in the kernel of T

k−1,lk−1,l−1 and

Tk,l−1k−1,l−1 respectively. Note that by Proposition 4, both the left- and right-hand

side of (11) are polynomials in Kerh−2Qk−1,l−1.Demanding that CC-II (ii) is satisfied, leads to the stated relation between

g1 and g2:

∂v∂x(ug1) = −∂v∂x

(

v − u〈v, ∂u〉k − l + 1

)

g2

⇔ − 2u〈∂v, ∂x〉g1 = 2v〈∂v, ∂x〉g2 − (m + 2l − 6)∂xg2

− 2

k − l + 1u〈∂u, ∂x〉g2 − 2

u〈v, ∂u〉〈∂v, ∂x〉k − l + 1

g2

⇔ − 2u〈∂v, ∂x〉g1 = −2(m + 2l − 4)(k − l + 2)

(m + 2k − 2)(k − l + 1)

× u[

〈∂u, ∂x〉 + 〈v, ∂u〉〈∂v, ∂x〉k − l + 2

]

g2

⇔ Tk−1,lk−1,l−1g1 = (m + 2l − 4)(k − l + 2)

(m + 2k − 2)(k − l + 1)T

k,l−1k−1,l−1g2,

where we have used explicitly that Qk,l−1g2 = 0. For g1 and g2 satisfying thisrelation, further calculations then show that CC-II (i) holds as well, i.e.,

∂u∂x

[

ug1 +(

v − u〈v, ∂u〉k − l + 1

)

g2

]

= 0.


Summarizing, the type B solutions of the operator Qk,l can be of the followingtype:

(i) choosing g2 = 0, we have that Kerh−1Qk−1,l ∩ Ker Tk−1,lk−1,l−1 ⊂ KerhQk,l;

(ii) choosing g1 = 0, we have that Kerh−1Qk,l−1 ∩ Ker Tk,l−1k−1,l−1 ⊂ KerhQk,l;

(iii) finally, choosing both g1 and g2 �= 0 is only possible if relation (11) is sat-isfied, which amounts to saying that certain elements in Kerh−2Qk−1,l−1

can also be inverted. This behaviour is different from what was obtainedfor the classical Rarita–Schwinger case, and is of course expected to holdin the more general case too. These summands could be described asIm T

k,l−1k−1,l−1 ∩ Im T

k−1,lk−1,l−1 ⊂ Kerh−2Qk−1,l−1.

In the special case k = l, there exists only one twistor operator and (11) reducesto

Tk,k−1k−1,k−1g2 = 0.

The type B solutions of the operator Qk,k are thus equivalent with elements of

Kerh−1Qk,l−1 ∩ Ker Tk,k−1k−1,k−1.

In both cases, we have thus obtained an inductive procedure to describe (atleast formally) the space of polynomial solutions for the operator Qk,l. In[2], we have proved this using a dimensional analysis, while in [3] we haveconstructed the explicit embedding factors realizing the decomposition of thekernel.

References

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tational algebra. In: Progress in Mathematical Physics, vol. 39. Birkhäuser, Basel (2004)10. Constales, D., Sommen. F., Van Lancker, P.: Models for irreducible representations of

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Kluwer, Dordrecht (1992)

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Essential Self-adjointness for CombinatorialSchrödinger Operators II-Metricallynon Complete Graphs

Yves Colin de Verdière · Nabila Torki-Hamza ·Françoise Truc

Received: 2 July 2010 / Accepted: 31 October 2010 / Published online: 20 November 2010© Springer Science+Business Media B.V. 2010

Abstract We consider weighted graphs, we equip them with a metric structuregiven by a weighted distance, and we discuss essential self-adjointness forweighted graph Laplacians and Schrödinger operators in the metrically noncomplete case.

Keywords Metrically non complete graph · Weighted graph Laplacian ·Schrödinger operator · Essential selfadjointness

Mathematics Subject Classifications (2010) 05C63 · 05C50 · 05C12 · 35J10 ·47B25

1 Introduction

This paper is a continuation of [21] which contains some statements aboutessential self-adjointness of Schrödinger operators on graphs. In [21], it was

Y. Colin de Verdière · F. Truc (B)Institut Fourier, Grenoble University, Unité mixte de recherche CNRS-UJF 5582, BP 74,38402 Saint Martin d’Hères Cedex, Francee-mail: [email protected]: http://www-fourier.ujf-grenoble.fr/∼trucfr/

Y. Colin de Verdièree-mail: [email protected]: http://www-fourier.ujf-grenoble.fr/∼ycolver/

N. Torki-HamzaFaculté des Sciences de Bizerte, Mathématiques et Applications (05/UR/15-02),Université du 7 Novembre à Carthage, 7021 Bizerte, Tunisiee-mail: [email protected], [email protected]

22 Y. Colin de Verdière et al.

proved that on any metrically complete weighted graph with bounded degree,the Laplacian is essentially self-adjoint and the same holds for the Schrödingeroperator provided the associated quadratic form is bounded from below. Theseresults remind those in the context of Riemannian manifold in [17] and alsoin [2, 19, 20]. There are many recent independent researches in locally finitegraphs investigating essential self-adjointness (see [7, 10–12, 16]), and relationsbetween stochastic completeness and essential self-adjointness (see [22, 24] aswell as the thesis [23]). Similar results have been extended for arbitrary regularDirichlet forms on discrete sets in [14] which is mostly a survey of the originalarticle [13]. More recently the paper [9] is devoted to the stability of stochasticincompleteness, in almost the same setup as in [13].

Here, we will investigate essential self-adjointness mainly on metrically noncomplete locally finite graphs.

Let us recall that a weighted graph G is a generalization of an electricalnetwork where the set of vertices and the set of edges are respectivelyweighted with positive functions ω and c. For any given positive function p,a weighted distance dp can be defined on G. Thus we have the usual notion ofcompleteness for G as a metric space.

The main result of Section 3 states that the weighted graph Laplacian �ω,c

(see the definition (1) below) is not essentially self-adjoint if the graph is offinite volume and metrically non complete (here the metric dp is defined using

the weights px,y = c− 1

2x,y ). The proof is derived from the existence of the solution

for a Dirichlet problem at infinity, established in Section 2.In Section 4, we establish some conditions implying essential self-

adjointness. More precisely, defining the metric dp with respect to the weights

px,y given by px,y = (min{ωx, ωy})c− 12

x,y, and addressing the case of metricallynon complete graphs, we get the essential self-adjointness of �ω,c + W pro-vided the potential W is bounded from below by N/2D2 , where N is themaximal degree and D the distance to the boundary. We use for this resulta technical tool deduced from Agmon-type estimates and inspired by the nicepaper [15], see also [5].

We discuss in Section 5 the case of star-like graphs. Under some assumptionson a, we prove that for any potential W, �1,a + W is essentially self-adjoint us-ing an extension of Weyl’s theory to the discrete case. In the particular case ofthe graph N, the same result had been proved in [1] (p. 504) in the contextof Jacobi matrices. We give some examples in Subsection 5.3 to illustrate thelinks between the previous results. Moreover we establish the sharpness of theconditions of Theorem 4.2.

The last Section is devoted to Appendix A dealing with Weyl’s limit point-limit circle criteria (see [18]) in the discrete case as well as in the continuouscase, and to Appendix B including the unitary equivalence between Laplaciansand Schrödinger operators [21] used repeatedly in Subsection 5.3.

Let us start with some definitions.G = (V, E) will denote an inf inite graph, with V = V(G) the set of vertices

and E = E(G) the set of edges. We write x ∼ y for {x, y} ∈ E.

Essential Self-adjointness for Combinatorial Schrödinger Operators II 23

The graph G is always assumed to be locally f inite, that is any x ∈ V hasa finite number of neighbors, which we call the degree (or valency) of x. Ifthe degree is bounded independently of x in V, we say that the graph G is ofbounded degree.

The space of real functions on the graph G is denoted

C(V) = { f : V −→ R}and C0(V) is the subspace of functions with finite support.

We consider, for any weight ω : V −→]0, +∞[, the space

l2ω(V) =

{f ∈ C(V);

∑x∈V

ω2x f 2(x) < ∞

}.

It is a Hilbert space when equipped with the inner product:

〈 f, g〉l2ω

=∑x∈V

ω2x f (x) .g (x) .

For any ω : V −→]0, +∞[, and c : E −→]0, +∞[, the weighted graph Lapla-cian �ω,c on the graph G weighted by the conductance c on the edges and bythe weigth ω on the vertices, is defined by:

(�ω,c f

)(x) = 1

ω2x

∑y∼x

cx,y ( f (x) − f (y)) (1)

for any f ∈ C(V) and any x ∈ V. If ω ≡ 1, we have(�1,a f

)(x) =

∑y∼x

ax,y ( f (x) − f (y)) .

Definition 1.1 Let p : E −→]0, +∞[ be given, the weighted distancedp(� +∞) on the weighted graph G is defined by

dp(x, y) = infγ∈�x,y

L(γ )

where �x,y is the set of the paths γ : x1 = x, x2, · · · , xn = y from x to y. Thelength L(γ ) is computed as the sum of the p-weights for the edges of thepath γ :

L(γ ) =∑

1�i�n

pxi,xi+1 .

In particular, if x and y are in distinct connected components of G, dp(x, y) =∞. We say that the metric space (G, dp) is complete when every Cauchysequence of vertices has a limit in V.

Definition 1.2 We denote by V the metric completion of (G, dp) and by V∞ =V \ V the metric boundary of V.


Definition 1.3 If G is a non finite graph and G0 a finite sub-graph of G, theends of G relatively to G0 are the non finite connected components of G \ G0.

2 The Dirichlet Problem at Infinity

We will use in this section the distance dp defined using the weights px,y = c− 1

2x,y .

Let us consider the quadratic form

Q( f ) =∑

{x,y}∈E

cx,y( f (x) − f (y))2 +∑x∈V

ω2x f (x)2,

which is formally associated to the operator �ω,c + Id on l2ω. We will need the

following result which is close to Lemma 2.5 in [12]:

Lemma 2.1 For any f : V → R so that Q( f ) < ∞ and for any a, b ∈ V, wehave

| f (a) − f (b)| �√

Q( f )dp(a, b).

Proof For any {x, y} ∈ E, | f (x) − f (y)| �√

Q( f )/√

cx,y. For any path γ froma to b , defined by the vertices x1 = a, x2, · · · , xn = b , we have | f (a) − f (b)| �√

Q( f )L(γ ). Taking the infimum of the righthandside with respect to γ we getthe result. �

Remark 2.1 Lemma 2.1 implies that any function f with Q( f ) < ∞ extends toV as a Lipschitz function f . We will denote by f∞ the restriction of f to V∞.

Theorem 2.1 Let us assume that (V, dp) is non complete. Let f : V → R withQ( f ) < ∞, then there exists a continuous function F : V → R which satisf iesboth conditions:

(i) (F − f )∞ = 0(ii) (�ω,c + 1)(F|V) = 0.

Moreover, such an F satisf ies Q(F) < ∞ and F ∈ l2ω.

If V is compact, such an F is unique.

Proof We will denote by A f the affine space of continuous functions G : V →R which satisfy Q(G) < ∞ and (G − f )∞ = 0.

Q is lower semi-continuous for the pointwise convergence on V as definedby Q = sup Qα with Qα( f ) = sum of a finite number of terms in Q.

Let Q0 := infG∈A f Q(G) and Gn be a corresponding minimizing sequence.The Gn’s are equicontinuous and pointwise bounded. From Ascoli’s Theorem,this implies the existence of a locally uniformly convergent subsequenceGnk → F. Using semi-continuity, we have Q(F) = Q0.


If x ∈ V and δx is the Dirac function at the vertex x, we have

ddt |t=0

Q(F + tδx) = 2ω2x[(�ω,c + 1)F(x)]

and this is equal to 0, because F is a minimum of Q restricted to A f .Uniqueness is proved using a maximum principle: let us assume that there

exists a non zero continuous F with F∞ = 0, then, changing, if necessary, Finto −F, there exists x0 ∈ V with F(x0) = maxx∈V F(x) > 0. The identity (ii)evaluated at the vertex x0 gives a contradiction. �

3 Not Essentially Self-adjoint Laplacians

Theorem 3.1 Let �ω,c be a weighted graph Laplacian and assume the followingconditions:

(i) (G, dp) with px,y = c− 1

2x,y is NON complete,

(ii) there exists a function f : V → R with Q( f ) < ∞ and f∞ �= 0.

Then �ω,c is not essentially self-adjoint.

Proof Because �ω,c is � 0 on C0(V), it is enough (see Theorem X.26 [18]) tobuild a non zero function F : V → R which is in l2

ω(V) and satisfies

(�ω,c + 1)F = 0. (2)

The function F given by Theorem 2.1 will be the solution of (2) the limit ofwhich at infinity is f∞. �

Remark 3.1 The assumptions of Theorem 3.1 are satisfied if (G, dp) is noncomplete and

∑ω2

y < ∞: it is enough to take f ≡ 1.They are already satisfied if G has a non complete “end” of finite volume.

Remark 3.2 Theorem 3.1 is not valid for the Riemannian Laplacian: if X is aclosed Riemannian manifold of dimension � 4, x0 ∈ X and Y = X \ x0, theLaplace operator on Y is essentially self-adjoint (see [4]) and Y has finitevolume.

Question 3.1 In Theorem 3.1, what is the def iciency index of �ω,c in terms ofthe geometry of the weighted graph?

4 Schrödinger Operators for Metrically non Complete Graphs

We now discuss essential selfadjointness for Schrödinger operators of thetype H = �ω,c + W on a graph G in the following setup: we define αx,y =min{ωx, ωy} and we assume that (G, dp), with px,y = αx,y√

cx,y, is non complete as

a metric space. It means that there exist Cauchy sequences of vertices without


limit in the set V. We will assume that G is of bounded degree, and we denotethe upper bound by N.

Definition 4.1 For a vertex x ∈ V, we denote by D(x) the distance to theboundary V∞ defined by

D(x) = infz∈V∞

dp(x, z).

Lemma 4.1 We have, for any edge {x, y},

|D(x) − D(y)| � dp(x, y) � min{ωx, ωy}√cx,y

. (3)

4.1 Agmon-type Estimates

Lemma 4.2 Let v, f ∈ C0(V) be real valued and assume Hv = 0. Then

〈 fv , H( fv)〉l2ω

=∑

{x,y}∈E

cx,yv(x)v(y)( f (x) − f (y))2. (4)

Proof In the case of positive v this type of formula is known as ground statetransform (see [8] and references within). A particular case of this computation(for operators of the type �1,a + W) can be found in [21], let us recall the prooffor the reader’s convenience:


=∑x∈V

f (x)v(x)

⎛⎝∑

y∼x

cx,y( f (x) − f (y))v(y)

⎞⎠

where we used the fact that Hv(x) = 0. An edge {x, y} contributes to the sumtwice. The total contribution is

f (x)v(x) cx,y( f (x) − f (y))v(y) + f (y)v(y)cy,x( f (y) − f (x))v(x)

so


=∑

{x,y}∈E

cx,y( f (x) − f (y)) ( f (x)v(x)v(y) − f (y)v(y)v(x)) .

�

Theorem 4.1 Let v be a solution of (H − λ)v = 0. Assume that v belongs tol2ω(V) and that there exists a constant c > 0 such that, for all u ∈ C0(V),

〈u|(H − λ)u〉l2ω

� N2

∑x∈V

max

(1

D(x)2, 1

)ω2

x|u(x)|2 + c‖u‖2l2ω, (5)

then v ≡ 0.

Proof This theorem is based on Lemma 4.2 applied to H − λ. Let us considerρ and R satisfying 0 < ρ < 1

2 and 1 < R < +∞. For any ε > 0, we define the


Fig. 1 The function F

1ε ρ R R + 1

F (u)

u

1

0

function fε : V → R by fε = Fε(D) where D denotes the distance associatedto the metric dp, and Fε : R

+ → R is the continuous piecewise affine functiondefined by (Fig. 1)

Fε(u) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

0 for u � ε

ρ(u − ε)/(ρ − ε) for ε � u � ρ

u for ρ � u � 11 for 1 � u � RR + 1 − u for R � u � R + 10 for u � R + 1

Using Lemma 4.2, Lemma 4.1 and the inequalities

v(x)v(y) � 1

2

(v(x)2 + v(y)2

),

the lefthandside of (4) is bounded as follows

〈 fεv|(H − λ)( fεv)〉l2ω

� 1

2

∑x∈V

v(x)2ω2x�ε(x),

with

�ε(x) =∑y∼x

( fε(x) − fε(y))2

dp(x, y)2� Nρ2

(ρ − ε)2

where the second inequality uses the fact that fε is ρ

ρ−ε−Lipshitz. This implies


� Nρ2

2(ρ − ε)2‖v‖2

l2ω. (6)

On the other hand, due to assumption (5) the lefthandside of (4) is boundedfrom below as follows:


� N2

∑ρ�D(x)�R

ω2xv(x)2 + c‖ fεv‖2

l2ω. (7)

Putting together (6) and (7) we get

N2

∑ρ�D(x)�R

ω2xv(x)2 + c‖ fεv‖2

l2ω

� Nρ2

2(ρ − ε)2‖v‖2

l2ω

. (8)

Then we do ε → 0. After that, we do ρ → 0 and R → ∞. We get v ≡ 0. �


Remark 4.1 The previous result is inspired by a nice idea from [15], so follow-ing the terminology of [15] we call (8) Agmon-type estimates.

4.2 Essential Self-adjointness

Theorem 4.2 Consider the Schrödinger operator H = �ω,c + W on a graphG, def ine αx,y = min{ωx, ωy} and assume that (G, dp), with pxy = αx,y√

cx,y, is non

complete as a metric space. For a vertex x ∈ V, we denote by D(x) the distancefrom x to the boundary V∞. We assume the following conditions:

(i) G is of bounded degree and we denote the upper bound by N,(ii) there exists M < ∞ so that

∀x ∈ V, W(x) � N2D(x)2

− M. (9)

Then the Schrödinger operator H is essentially self-adjoint.

Remark 4.2 In the particular case when∑

ω2x < ∞, the Laplacian H = �ω,c

does not satisfy the assumption (9) so this result is coherent with Theorem 3.1.

Remark 4.3 The exponent of D(x) in (9) is sharp. In fact, one can find apotential W such that W(x) � k

D(x)2 where k < N2 and weights ω and c such

that H = �ω,c + W is non essentially self-adjoint. See Example 5.3.2.

Remark 4.4 In the case where ω ≡ 1 the result is an immediate consequenceof [13] (Theorem 5).

Proof We have, for any u ∈ C0(V)

〈u|Hu〉l2ω

�∑x∈V

W(x)ω2x|u(x)|2,

so using assumption (9) we get:

〈u|(H − λ)u〉l2ω− N

2

∑x∈V

1

D(x)2ω2

x|u(x)|2 �∑x∈V

−(M + λ)‖u‖2l2ω

.

Then choosing for example

λ = −M − 1

we get the inequality (5) with c = 1, and the proof follows from Theorem 4.1.�


5 Schrödinger Operators on “Star-like” Graphs

5.1 Introduction

Definition 5.1 The graph N is the graph defined by V = {0, 1, 2, · · · } and E ={{n, n + 1} | n = 0, 1, · · · }.

Definition 5.2 We will call an infinite graph G = (V, E) star-like if there existsa finite sub-graph G0 of G so that G \ G0 is the union of a finite number ofdisjoint copies Gα of the graph N (the ends of G relatively to G0 according toDefinition 1.3).

For example, the graph Z, defined similarly to N, is star-like.Let us consider a Laplace operator L = �1,a on G. On each end Gα of G, L

will be given by

Lα fn = −aαn,n+1 fn+1 + (

aαn−1,n + aα

n,n+1

)fn − aα

n−1,n fn−1 ,

where the aαn−1,n’s are > 0. If W : V → R, we will consider Schrödinger opera-

tors H on C0(G) defined by H = �1,a + W.

Lemma 5.1 Let G0 be a f inite sub-graph of G. The operator H = �1,a + W onG is essentially self-adjoint if and only if it is essentially self-adjoint on each endof G relatively to G0. More precisely, the def iciency indices n± are the sum ofthe corresponding def iciency indices of the ends.

We will need the following Lemma which is a consequence of Kato-RellichTheorem, see [7], Proposition 2.1:

Lemma 5.2 If A and B are 2 symmetric operators with the same domains andR = B − A is bounded, then the def iciency indices of A and B are the same.

Proof We give here an alternative proof to this result. Let us define, fort ∈ R, At = A + tR so that A0 = A and A1 = B. The domains of the closuresof the At’s coïncide: the “graph-norms” ‖Atu‖l2 + ‖u‖l2 are equivalent. Thedomains of the adjoints coïncide too. Let K = D(A�)/D(A) and Qt(u, v) =−i

(〈A�t u|v〉 − 〈u|(A�

t v〉) which is well defined on K. We know that thesebounded Hermitian forms are non degenerate on K with the graph normand continuous w.r. to t. Hence the Morse index n−(t) is locally constant:take a decomposition K = K+ ⊕ K− where q = Qt0 satisfies q|K+ � C > 0 andq|K− � −C < 0. �

Using Lemma 5.2, we can prove Lemma 5.1:

Proof We will consider the operator Hred where we replace the entries ax,y ofH with {x, y} ∈ E(G0) by 0. The claim of the Lemma is clear for Hred becauseit is the direct orthogonal sum of the Schrödinger operators of the ends and a


finite rank l2−bounded matrix. We can then use Lemma 5.2 because H − Hred

is bounded. �

Remark 5.1 It follows from Lemma 5.1 that, concerning essential self-adjointness questions for star-like graphs, it is enough to work on the graphN. We have

(H f )0 = −a0,1 f1 + a0,1 f0 + W0 f0.

This implies that the space of solutions of (H − λ)u = 0 on N is of dimension1 and any solution so that f0 vanishes is ≡ 0. We will consider also solutions“near infinity”, i.e. ( fn)n�0 satisfies ((H − λ) f )n = 0 for n � 1; this space is ofdimension 2.

5.2 Main Result

It is known [6] that H = �1,a + W is essentially self-adjoint provided �1,a isbounded as an operator on l2(G) and W bounded from below. For star-likegraphs, we have the following result, which holds for any potential W:

Theorem 5.1 If G is star-like and if for each end Gα ,

1/aαn−1,n /∈ l1(N) (10)

then H = �1,a + W with domain C0(V) is essentially self-adjoint for anypotential W.

Remark 5.2 The condition (10) is sufficient but not necessary. See Example5.3.2.

Proof Due to Remark 5.1 we only have to prove the following

Theorem 5.2 If

1

an−1,n/∈ l1(N), (11)

the Schrödinger operator H = �1,a + W with domain C0(N) is essentially self-adjoint for any potential W.

This result is contained in the book [1] (p. 504). We propose here a shortproof, obtained by contradiction using Corollary 6.1 which is an analog ofWeyl’s limit point-limit circle criteria in the discrete case.

Let us consider an operator �1,a such that (10) is fulfilled. We assume thatany sequence u, such that (H − i)u = 0 near infinity, is in l2(N). In particular,there exists a basis f, g of solutions of (H − i) f = 0 with f ∈ l2(N) and g ∈l2(N).

We have

−an,n+1 fn+1 + (an−1,n + an,n+1 + (Wn − i)) fn − an−1,n fn−1 = 0 ,


and the same holds for g. The Wronskian of f and g is the sequence Wn =fngn−1 − fn−1gn. We have, for any n ∈ N:

Wn+1 = an−1,n

an,n+1Wn ,

which implies

Wn = a0,1

an−1,nW1 .

But since the Wronskian is in l1(N) according to the assumption that f and gare in l2(N), we get a contradiction with the hypothesis (10). �

5.3 Examples of Schrödinger Operators

5.3.1 Example 1

Let us consider the Laplacian �ω,c on N, with, ∀n > 0, cn−1,n = n3 and, ∀n � 0,ωn = 1

n+1 . Since∑

ω2n < ∞ and

∑c−1/2

n−1,n < ∞ we deduce from Theorem 3.1(due to Remark 3.1) that �ω,c is not essentially self-adjoint.

Applying a result of [21] (see Proposition 7.1 in Appendix B) we get that thisLaplacian is unitarily equivalent to the Schrödinger operator H = �1,a + Wwith an−1,n = cn−1,n

ωn−1ωn∼ n5 and

Wn = 1

ωn

[cn,n+1

(1

ωn− 1

ωn+1

)+ cn−1,n

(1

ωn− 1

ωn−1

)]∼ −3n3,

which is therefore not essentially self-adjoint.According to Theorem 5.1, such an operator must verify 1

an−1,n∈ l1(N), which

is indeed the case.

5.3.2 Example 2: Discretization of a Schrödinger Operator on R+

Let us consider the Schrödinger operator on ]0, +∞[ defined on smooth com-pactly supported functions by Lf := − f " + A

x2 f . This operator is essentiallyself-adjoint if and only if A > 3/4 (see [18] Theorem X 10). We discretize thisoperator in the following way: let us consider the graph � = (V, E) resultingof the following dyadic subdivision of the interval (0, 1): the vertices are thexn = 2−n and the edges are the pairs {2−n, 2−n+1} which correspond to theintervals [2−n, 2−n+1] of length ω2

n = 2−n.Then we define, for any

f ∈ l2ω(V) =

{f ∈ C(V) |

∑n∈N

2−n f 2n < +∞

}

where we set f = ( fn), the quadratic form

Q( f ) =∑n∈N

2−n

[(fn+1 − fn

2−n

)2

+ A22n f 2n

].


According to the previous definitions and if we set cn,n+1 = 2n, this quadraticform is associated to the Schrödinger operator H = �ω,c + W on N with thepotential Wn := A22n.

Let us set an,n+1 = cn,n+1

ωnωn+1= 22n+ 1

2 . Applying Proposition 7.1 we get that H isunitarily equivalent to

H = �1,a + W + W

with

Wn = 1

ωn

[cn,n+1

(1

ωn− 1

ωn+1

)+ cn−1,n

(1

ωn− 1

ωn−1

)]= 22n

(3

2− 5

√2

4

).

We have H = �1,a + (A − A0)4n with A0 = 5√

24 − 3

2 (> 0). The metric graph(N, dp) with pn,n+1 = a−1/2

n,n+1 is non complete. The solutions u of Hu = 0 verify

4un+1 −(

5 + 2√

2(A − A0))

un + un−1 = 0.

The solutions are generated by αn1 and αn

2 where α1 and α2 are the roots of

4α2 −(

5 + 2√

2(A − A0))

α + 1 = 0.

We have |α1| < 1 and |α2| < 1 if and only if A0 − 5√2

< A < A0.

Using Proposition 6.1, with d = 2 and Un =(

un

un−1

), we get, for any λ ∈

C, the exponential decay of all solutions near infinity of (H − λ)u = 0 ifA0 − 5√

2< A < A0, and the existence of a solution of (H − λ)u = 0 with

exponential growth in the case when A > 5√

24 − 3

2 or A < − 5√

24 − 3

2 .Hence (by Corollary 6.1) we get the following result:

Proposition 5.1

1. If − 5√

24 − 3

2 < A < 5√

24 − 3

2 , then the discretized operator H is not essen-tially self-adjoint.

2. If A > 5√

24 − 3

2 (�) or A < − 5√

24 − 3

2 , then H is essentially self-adjoint.

From this result we can deduce several informations:

1. The condition (�) is analogous to the condition A > 3/4 in the continuouscase.

2. Proposition 5.1 implies that for A = 0 the operator H = �ω,c is notessentially self-adjoint, which is a result predicted by Theorem 3.1.

3. This gives examples of essentially self-adjoint operators with 1/an ∈ l1.4. Sharpness of the assumption (9) in Theorem 4.2

In this context, the distance dp is associated to

px,y = αx,y√cx,y


with αx,y = min{ωx, ωy} so we get

D(n) =∑p�n

αp,p+1√cp,p+1

=∑p�n

(2−p−1

2p

)1/2

= 2− 12 2−n2

so1

D(n)2= 22n−1 .

If the operator H = �ω,c + A4n satisfies the assumption (9), thenA > 1

2 which involves condition (�), since 12 > 5

√2

4 − 32 , so Theorem 4.2

is coherent with proposition 5.1. Moreover the operator H = �ω,c + A4n

with A = 5√

24 − 3

2 is not essentially self-adjoint, which implies that theestimate (9) on the growth of the potential in Theorem 4.2 is sharp.

5.3.3 Example 3

Let us consider the Laplacian �ω,c on N, where the coefficients verify cn−1,n =nγ with γ > 2 and ωn = (n + 1)−β with β > 1

2 . Since∑

ω2n < ∞ and

∑c−1/2

n−1,n <

∞ we deduce from Theorem 3.1 (due to Remark 3.1) that �ω,c is not essentiallyself-adjoint.

Applying one more time Proposition 7.1, we see that this operator isunitarily equivalent to the Schrödinger operator H = �1,a + W, with an−1,n ∼nγ+2β and the potential Wn ∼ −β(β + γ − 1)n2β+γ−2, which is therefore alsonot essentially self-adjoint. We emphasize that W is not bounded from below,which is predicted in [21], Theorem 3.2.

Furthermore, according to Theorem 5.1, such an operator must verify thecondition 1

an−1,n∈ l1(N), which is indeed the case. Following the terminology

of the previous sections, it means the non completeness of (N, dp) with theweights pn−1,n = a−1/2

n−1,n.

5.3.4 Example 4

Let us consider the Laplacian H = �ω,c on a spherically homogeneous rootedtree G = (V, E) (see [3] and references within). For any vertex x, we denoteby δ(x) the distance from x to the root 0 and define ωx = 2−δ(x), and cx,y = 2δ(x),for any y ∼ x so that δ(y) = n + 1 . We assume that the graph G has a uniformdegree N + 1.

Let us set ax,y = cx,y

ωxωy. We have ax,y = 23n+1 for any edge x, y, so that δ(x) =

n and δ(y) = n + 1. Then, due to Proposition 7.1, the operator H is unitarilyequivalent to

H = �1,a + W

with

W(x) = 23n(

−N + 1

4

)for any x such that δ(x) = n .


The radial solutions u of Hu = 0 can be seen as sequences (un) which satisfythe equation:

−2Nun+1 +(

N + 1

2

)un − 1

4un−1 = 0 .

The solutions are generated by αn1 and αn

2 where α1 and α2 are the roots of

α2 −(

1

2− 1

4N

)α + 1

8N= 0.

We have |α1| < 1 and |α2| < 1 for any N > 0.The radial solutions of (H − λ)u = 0 satisfy

−2Nun+1 +(

N + 1

2

)un − 1

4un−1 = 2Nλ2−(3n+1)un .

Using Proposition 6.1, with d = 2 and Un =(

un

un−1

), we get the exponential

decay of all solutions near infinity of (H − λ)u = 0.Hence (by Corollary 6.1) we get the following result:

Proposition 5.2 For any N � 1 H is not essentially self-adjoint.

Remark 5.3 We have

∑x

ω2x =

∑n

ω2n Nn =

∑n

(N4

)n

.

If N < 4, then∑

x ω2x < ∞ so Theorem 3.1 can also be applied to get the result

since the graph is non complete with respect to the metric dp , with px,y = c− 1

2x,y .

Acknowledgements The second author is greatly indebted to the research unity“Mathématiques et Applications” (05/UR/15-02) of Faculté des Sciences de Bizerte (Tunisie) forthe financial support, and would like to present special thanks to Institut Fourier where this workwas carried on.

Thanks to D. Lenz for giving notes on some references.All the authors would like to thank the anonymous referee for careful reading, numerous

remarks, useful suggestions and valuable references.

Appendix A: Weyl’s “Limit Point-limit Circle” Criteria

The Discrete Case

The goal of this section is to prove the discrete version of the Weyl’s “limitpoint-limit circle” criterium. Our presentation is simpler than the classicalpresentation for the continuous case (see [18], Appendix to Section X.1).


Let us consider the Hilbert space H := l2(N, CN) and the formally symmet-

ric differential operator P defined by

Pf (0) = P0,0 f (0) + P0,1 f (1), ∀l � 1,

Pf (l) = Pl,l−1 f (l − 1) + Pl,l f (l) + Pl,l+1 f (l + 1)

where

1. ∀l � 1, P�l−1,l = Pl,l−1

2. ∀l � 0, P�l,l = Pl,l

3. ∀l � 0, Pl,l+1 is invertible4. ∃M ∈ R so that for any f ∈ C0(N, C

N), QP( f ) = 〈Pf | f 〉 � −M‖ f‖2.

Let us define the subspace E of H as the set of l2 sequences f so that, forall l � 1, (P − i) f (l) = 0; the space E is isomorphic to the space of germs atinfinity of l2 solutions of (P − i) f = 0. Assumption 3. implies that dim E � 2N.Let us denote by K = ker(P − i) ∩ l2 and consider the following sequence

0 → K → E → CN → 0, (12)

where the non trivial arrow is given by f → (P − i) f (0). We have the

Theorem 5.3 The sequence (12) is exact and the def iciency indices n± = dimKof P are given by n± = dim E − N.

Proof Assumption 4. implies (using Corollary of Theorem X.1 in [18]) thatthe deficiency indices are equal. The only non trivial point is to prove that thearrow p : E → C

N is surjective. Let us consider P a self-adjoint extension of Pwhich exists because n+ = n−. Let us consider the map ρ : C

N → E defined byρ(x) = (P − i)−1(x, 0, 0, · · · ). Then p ◦ ρ = IdCN . �

Corollary 5.1 The Schrödinger operator H = �1,a + W def ined on C0(N) isessentially self-adjoint if and only if there exists a sequence u such that (H −i)u = 0 near inf inity (i.e. ((H − i)u)n = 0 for n large enough) which is not inl2(N).

Asymptotic Behavior of Perturbed Hyperbolic Iterations

In order to apply Corollary 5.1, the following results will be useful

Proposition 5.3 Let us consider the following linear dynamical system on Cd:

∀n � 0, Un+1 = AUn + R(n)Un (13)

where

1. A is hyperbolic: all eigenvalues λ j of A satisfy |λ j| �= 12. R(n) → 0 as n → ∞.


Then

• Case A: If all eigenvalues λ j of A satisfy |λ j| < 1, all solutions (Un) of (13)are exponentially decaying.

• Case B: If m eigenvalues satisfy |λ j| > 1, then there exists an m-dimensionalvector space F of solutions of (13) whose non-zero vectors have exponentialgrowth.

Proof

Case A There exists a norm ‖.‖ on Cd so that the operator norm of A satisfies

‖A‖ = k < 1. For n large enough, we have ‖A + R(n)‖ � k′ < 1. Theconclusion follows.

Case B There exists a splitting Cd = Y ⊕ Z , denoted x = y + z, with dim Y =

m, stable by A, norms on Y and Z and 2 constants μ < 1 < σ , so that

∀y ∈ Y, ‖Ay‖ � σ‖y‖,∀z ∈ Z , ‖Az‖ � μ‖z‖.

Let us choose ε > 0 so that 1 < σ − 2ε and N so that ‖R(n)‖ � ε forn � N. We have, for n � N,

‖yn+1‖ � σ‖yn‖ − ε(‖yn‖ + ‖zn‖), ‖zn+1‖ � μ‖zn‖ + ε(‖yn‖ + ‖zn‖),so that

‖yn+1‖ − ‖zn+1‖ � (σ − 2ε)(‖yn‖ − ‖zn‖).Any solution which satisfies ‖yN‖ > ‖zN‖ will have exponentialgrowth. Take for F the space of solutions for which zN = 0. �

The Continuous Case

A similar method works for the continuous case. Let H = − d 2

dx 2 + A(x) bea system of differential operators where A(x) is Hermitian for every x andis continuous on [0, a[ as a function of x. The differential operator H is L2-symmetric on the Dirichlet domain

D = C∞0

([0, a[, CN) ∩ {u | u(0) = 0}.

We denote HD the closure of (H, D). Let us assume that n+(HD) = n−(HD)

which is true for example if A is bounded from below or if A is real-valued.Then

Theorem 5.4 If E is the space of solutions u of the dif ferential equation (H − i)u = 0 which are L2 near a, then n±(HD) = dim E − N.

Proof Let us consider the sequence

0 → ker(HD − i) → E → CN → 0, (14)


where the only non trivial arrow is given by u → u(0). This sequence is exact:we have only to prove the surjectivity of the non trivial arrow. Let H be a self-adjoint extension of HD and χ ∈ C∞

0 ([0, a[, R) with χ(0) = 1. For any X ∈ CN ,

let us consider

u = χ X − (H − i)−1 ((H − i)(χ X)) .

Then (H − i)u = 0, u(0) = V and u is L2 near a. �

Appendix B: Unitary Equivalence Between Laplacians and SchrödingerOperators

In this section, we recall the following results (see [21] Proposition 2.1 andTheorem 5.1): the first one states that a Laplacian is always unitarily equivalentto a Schrödinger operator, and the second result asserts that a Schrödingeroperator with a strictly positive quadratic form is unitarily equivalent to aLaplacian.

For a weighted graph G by the weight ω on its vertices, let

Uω : l2ω (V) −→ l2 (V)

the unitary operator defined by

Uω ( f ) = ω f.

This operator preserves the set of functions on V with finite support.

Proposition 5.4 The operator

� = Uω �ω,c U−1ω ,

is a Schrödinger operator on G. More precisely:

� = �1,a + W

where a is a strictly positive weight on E given by:

ax,y = cx,y

ωxωy

and the potential W : V −→ R is given by:

W = − 1

ω�1,aω.

The following Theorem uses the existence of a strictly positive harmonicfunction (see [21], Section 4).

Theorem 5.5 Let P a Schrödinger operator on a graph G. We assume that〈Pf, f 〉l2 > 0 for any function f in C0 (V) \ {0}. Then there exist weights: ω onV and c on E such that P is unitarily equivalent to the Laplacian �ω,c on thegraph G.


References

1. Berezans’kii, J.M.: Expansions in eigenfunctions of selfadjoint operators. Translations ofMathematical Monographs, vol. 17. American Mathematical Society, Providence (1968)

2. Braverman, M., Milatovic, O., Shubin, M.: Essential self-adjointness of Schrödinger-typeoperators on manifolds. Russ. Math. Surv. 57, 641–692 (2002)

3. Breuer, J.: Singular continuous spectrum for the Laplacian on certain sparse trees. Commun.Math. Phys. 269(3), 851–857 (2007)

4. Colin de Verdière, Y.: Pseudos-Laplaciens I. Ann. Inst. Fourier (Grenoble) 32, 275–286 (1982)5. Colin de Verdière, Y., Truc, F.: Confining quantum particles with a purely magnetic field. Ann.

Inst. Fourier (Grenoble) 60(5) (2010)6. Dodziuk, J.: Elliptic operators on infinite graphs. Analysis Geometry and Topology of Elliptic

Operators, pp. 353–368, World Sc. Publ., Hackensack (2006)7. Golénia, S., Schumacher, C.: The problem of deficiency indices for discrete Schrödinger

operators on locally finite graphs. arXiv:1005.0165 (2010)8. Haeseler, S., Keller, M.: Generalised solutions and spectrum for Dirichlet forms on graphs.

arXiv:1002.1040 (2010)9. Huang, X.: A note on stochastic incompletness for graphs and weak Omori-Yau maximum

principle. arXiv:1009.2579 (2010)10. Jorgensen, P.E.T.: Essential self-adjointness of the graph-Laplacian. J. Math. Phys. 49(7),

073510, 33 pp. (2008)11. Jorgensen, P.E.T., Pearse, E.P.J.: Spectral reciprocity and matrix representations of un-

bounded operators. arXiv:0911.0185 (2009)12. Jorgensen, P.E.T., Pearse, E.P.J.: A discrete Gauss-Green identity for unbounded Laplace

operators, and the transience of Random walks. arXiv:0906.1586 (2010)13. Keller, M., Lenz, D.: Dirichlet forms and stochastic completneness of graphs and subgraphs.

arXiv:0904.2985 (2009)14. Keller, M., Lenz, D.: Unbounded Laplacians on graphs: basic spectral properties and the heat

equation. Math. Nat. Phenomena 5(4), 198–224 (2010)15. Nenciu, G., Nenciu, I.: On confining potentials and essential self-adjointness for Schrödinger

operators on bounded domains in Rn. Ann. Henri Poincaré 10, 377–394 (2009)

16. Masamune, J.: A Liouville property and its application to the Laplacian of an infinite graph.Contemp. Math. 484, 103–115 (2009)

17. Oleinik, I.M.: On the essential self-adjointness of the operator on complete Riemannianmanifolds. Math. Notes 54, 934–939 (1993)

18. Reed, M., Simon, B.: Methods of modern mathematical physics. II-Fourier Analysis, Self-adjointness. New York Academic Press (1975)

19. Shubin, M.: The essential self-adjointness for semi-bounded magnetic Schrödinger operatorson non-compact manifolds. J. Funct. Anal. 186, 92–116 (2001)

20. Shubin, M.: Classical and quantum completness for the Schrödinger operators on non-compactmanifolds. Geometric Aspects of Partial Differential Equations (Proc. Sympos., Roskilde,Denmark 1998), vol. 242, pp. 257–269. Amer. Math. Soc. Providence (1999)

21. Torki-Hamza, N.: Laplaciens de graphes infinis I Graphes métriquement complets.Confluentes Mathematici 2(3) (2010, to appear)

22. Weber, A.: Analysis of the physical Laplacian and the heat flow on a locally finite graph.J. Math. Anal. Appl. 370, 146–158 (2010)

23. Wojiechowski, R.K.: Stochastic Completeness of Graphs. Ph.D. Thesis, The Graduate Centerof the University of New York (2008)

24. Wojiechowski, R.K.: Heat kernel and essential spectrum of infinite graphs. Univ. Math. J.58(3), 1419–1442 (2009)

http://arxiv.org/abs/1005.0165







An Asymptotic Comparison of DifferentiableDynamics and Tropical Geometry

Tsuyoshi Kato

Received: 5 February 2010 / Accepted: 8 December 2010 / Published online: 5 January 2011© Springer Science+Business Media B.V. 2010

Abstract In this paper we introduce a new comparison method to give roughasymptotic estimates of different evolutional dynamics. It uses a kind ofscale transform called tropical geometry, which connects automata with realrational dynamics. By this procedure the defining equations are transformedrather than solutions themselves. Real rational dynamics is regarded as anapproximation of evolutional dynamics given by partial differential equations(PDEs). Two different evolutional dynamics can be considered when theirdefining equations are transformed to the same automata at infinity.

Keywords Large scale geometry · Tropical geometry ·Scale transform of dynamical systems

Mathematics Subject Classifications (2010) 35A30 · 37E05 · 39A14 · 39A22

1 Introduction

1.1 Asymptotic Comparison Between Solutions to Different PDEs

Scaling limits connect several dynamics whose features are often very differentmutually. One of particular properties of scaling limits is that in many casessuch associations are not injective. When two dynamical systems correspond tothe same one by such scaling limits, then one might say that these two dynamicsbehave by the same way at infinity, and so expect that they will hold somecommon structural similarity.

T. Kato (B)Department of Mathematics, Kyoto University, Kyoto 606-8502, Japane-mail: [email protected]

40 T. Kato

Motivated by such aspects, in this paper we study large scale analytic prop-erties of solutions to evolutional differential equations by use of a particulartype of scaling limit. It consists of two steps, where one is to associate discretedynamics given by real rational functions from differential equations, and thesecond is automata given by (max, +)-functions from the rational dynamics,which appears in tropical geometry. Combination of these two steps gives aprocess of association of automata from PDE. As above one of the importantobservations for the process is that it is not one to one, and so differentdifferential equations can correspond to the same dynamics by automata. Thesituation can be interpreted that large (also very small) valued solutions tothese PDEs admit mutual analytic relations in some sense, which we wouldexpect to lead us to large scale analysis of structure for classes of differentialequations.

In this paper we introduce a new method for study of solutions of nonlinear partial differential equations. Our main interest here is to obtain relativeestimates of asymptotic growth of solutions to different PDEs with respect tohigher derivatives and initial conditions.

For T0 ∈ (0, ∞], let u : (0, ∞) × [0, T0) → (0, ∞) be a function of classCα+1. Then we introduce uniform norm of u of order α + 1 by:

||u||α+1 = max∂i= ∂x, ∂s

{∥∥∥∥ ∂α+1u∂1 . . . ∂α+1

∥∥∥∥C0((0,∞)×[0,T0))

}.

Let c = inf(x,s)∈(0,∞)×[0,T0) u(x, s) � 0 be the infimum of u. Suppose u(x, s) �c > 0 is positive. Then we introduce the higher derivative rates by:

K(u) ≡ ||u||α+1

cand call them the derivative rates of order α + 1. Notice that even when c issufficiently large, still K can be small when functions u are ‘near’ polynomialof order less than α.

Let u, v : (0, ∞) × [0, T0) → (0, ∞) be two functions of class Cα+1. For smallε > 0, we introduce the initial rates:

[u : v]ε ≡ sup(x,s)∈(0,∞)×[0,εq]∪(0,ε p]×[0,T0)

(u(x, s)v(x, s)

)±1

.

Our method provides with asymptotic relative growth very explicitly forsolutions to different differential equations, with respect to their higher deriva-tive rates and initial rates. Let us consider two differential equations of order atmost α, P(u, ux, us, u2x, uxs, . . . ) = 0 and Q(v, vx, vs, v2x, vxs, . . . ) = 0, and takepositive solutions u, v : (0, ∞) × [0, T0) → (0, ∞) of class Cα+1. When both Pand Q are ‘induced from the same automaton ϕ’ which we clarify below, thenwe verify that there exist constants C = C(y, r, K) which depend only on the‘rough structure’ of the differential equations P and Q, which are independentof individual solutions, so that they satisfy uniform bounds:

u(x, s)v(x, s)

,v(x, s)u(x, s)

� C(x + ks, r, K)

Differentiable Dynamics and Tropical Geometry 41

when their higher derivative rates and intial rates satisfy bounds K(u), K(v) �K and [u : v](L+1)(2CK)−1 � r respectively. Here k, L = max(l, d) and C areexplicit constants which arise from scaling limits of these PDE as below.

Let us pick up the required information to determine the constantsC(y, r, K). Our basic process is to extract very rough framework of structuresof PDE. They are given by n variable rational dynamics of the form:

zt+1N+1 = f

(zt+1

N−l0, . . . , zt+1

N , ztN−l1

, . . . , ztN+k1

, . . . , zt−dN+kd+1

)and scaling parameters zt

N = εmu(x, s) and (N, t) = (ε−px, ε−qs).Once such reductions are given, then automata ϕ are canonically associated,

and at this stage, one has chosen several numbers L = max(l, d), k, n, D =max(p, q) and C, where l = max(l0, . . . , ld+1), k = max(k1, . . . , kd+1), and Care the coefficients of α + 1 derivatives in the Taylor expansions, called errorconstants (see ‘Higher Derivative Rates’ in Section 3.2.1). Relative (max, +)-functions ϕ are piecewise linear and they are Lipschitz. So one obtains particu-lar two data M and c, where M is the number of the components (Section 1.2)and c is the Lipschitz constants both for ϕ. In total at the level of definingequations of dynamics, induction of rational functions and scaling parametersdetermine the above seven data. In Section 4 we see that these constants areexplicitly calculated or estimated in concrete cases.

On the other hand individual solutions give the constants [u : v]ε and K,Now C(y, r, K) are in fact given quite explicitly as below. The above numbersare all the data which we need for the above asymptotic estimates amongapplicable pairs of PDEs.

As a general procedure, the rational dynamics with the scaling parametersabove give pairs of partial differential equations F(ε, u, ux, . . . ) = 0 as the lead-ing terms, and the error terms F1(ε, u, ux, . . . ) = 0 by use of Taylor expansions(Section 1.3).

Let us state our main theorem. The following comparison method discoversvery rough structural similarity among different partial differential equations:

Theorem 1.1 Let f and g be both relatively elementary and increasing functionsof n variables, which are mutually tropically equivalent. Let F and G be theirleading terms of order at most α � 0, and take positive Cα+1 solutions u, v :(0, ∞) × [0, T0) → (0, ∞) with:

F(ε, u, ux, us, . . . , uαx, uαs) = 0, G(ε, v, vx, vs, . . . , vαx, vαs) = 0.

Assume both u and v are ε0 controlled bounded by C. Then for any 0 < ε �min

(1

2C , ε0)

and D = max(p, q), the estimates hold:(u(x, s)v(x, s)

)±1

� (2M)8 cε−D(x+ks)+1−1

c−1 ([u : v](L+1)ε)cε−D(x+ks)+n

.

From this we will induce various estimates in concrete examples withrespect to their higher derivative rates below. We notice that as a general

42 T. Kato

principle, double exponential growth are optimal in our setting (remark (2)in Section 2.3).

Now what are the rest is to find suitable pairs of PDEs which arise fromthe same automata, or in other words, to find suitable rational functions whichproduce the desired PDEs. This is the key step for our general machinery ofdiscretization of PDE.

Let us see explicit estimates for concrete cases (see [8]). Here we treat twoequations, one is quasi linear equations of order 1, and the other is diffusionequations of order 2. The proofs contain two fundamental techniques, whereone is cancellation, and the other is linear deformation both for rationalfunctions. They are obtained by combinations of results in Section 4 withLemma 3.3.

Firstly let us consider the quasi linear equations, and choose the uniformnorm of second order:

||u||2 = max

{∥∥∥∥∂2u∂x2

∥∥∥∥C0

,

∥∥∥∥∂2u∂s2

∥∥∥∥C0

,

∥∥∥∥ ∂2u∂x∂s

∥∥∥∥C0

}

We put the second derivative rates K(u) = ||u||2inf(x,s)∈(0,∞)×[0,T0) u(x,s) .

Let us fix any positive constant K0 > 0.

Theorem 1.2 For any 0 < ε � 0.1K−10 , let v, u : (0, ∞) × [0, T0) → (0, ∞) be

C2 solutions to the quasi linear equations:

vs + εvvx − 1

2v2 = 0, 2us + εu(us + ux) = 0.

Suppose their second derivative rates are bounded by K0 � K(u), K(v). Thenthey satisfy the asymptotic estimates for all (x, s) ∈ (0, ∞) × [0, T0):(

u(x, s)v(x, s)

)±1

� 402ε−1(x+2s)+4([u : v]2ε)

2ε−1(x+2s)+3.

In particular when u(x, s) ≡ R > 0 is constant, then the estimates hold:

R(40)−2ε−1(x+2s)+4([v : R]2ε)

−2ε−1(x+2s)+3(1)

� v(x, s) � R(40)2ε−1(x+2s)+4([v : R]2ε)

2ε−1(x+2s)+3. (2)

Next we treat diffusion equations. Let F be an elementary and increasingfunction. Here we consider the diffusion equations of the type:

us = u2x + F(u).

There has been various studies for such type of diffusion equations, inrelation with blowing up of solutions. We point out two known results.

(1) Let F(u) = ul for l = 1, 2, . . . If l = 2, then any positive solutions to theequation blow up at finite time. For l � 4, it has global positive solutionsfor small initial values [1]. The number 3 is called the Fujita index (for onedimensional case).


(2) For all l, if the initial functions take sufficiently large values, then suchsolutions blow up at fintie time [6].

For this case we take the uniform norm of the third derivatives:

||u||3 = max

{∥∥∥∥∂3u∂x3

∥∥∥∥C0

,

∥∥∥∥∂3u∂s3

∥∥∥∥C0

,

∥∥∥∥ ∂3u∂x2∂s

∥∥∥∥C0

,

∥∥∥∥ ∂3u∂x∂s2

∥∥∥∥C0

}.

Then we put the third derivative rates K(u) ≡ ||u||3inf(x,s)∈(0,∞)×[0,T0) u(x,s) .

Firstly let us compare linear diffusion equations with advection-diffusionequations of variable exchange. For the linear case, the correspond-ing Lipschitz constant is equal to one, and one obtains the exponentialasymptotics:

Proposition 1.3 Let us f ix K0 > 0, and choose any 0 < ε � (200K0)−1. Let

u, v : (0, ∞) × [0, ∞) → (0, ∞) be C3 solutions to the linear equations:

7

5us − 193

40u2x = 0,

15

8εvs + 43

32vx − 19

16ε3v2s = 0.

Suppose the third derivative rates satisfy the bounds K(u), K(v) � K0. Thenthey satisfy the exponential asymptotic estimates for all (x, s) ∈ (0, ∞) × [0, ∞):(

u(x, s)v(x, s)

)±1

� 1048(ε−2(x+4s)+1)[u : v]5ε .

This is obtained by applying tropical linear deformation of rational func-tions. Such method is also applied for non linear case as below.

For 1 < a ∈ Q, let us consider the diffusion equations of the form:

us = u2x + ua.

Let us consider the special solution v : [0, S0) → (0, ∞) given by:

v(s) = c

(1 − ca−1(a − 1)s)(a−1)−1

where S0 = 1ca−1(a−1)

. Both v and its third derivative are increasing functions.Thus for any 0 < s0 < S0 and α = (a − 1)−1, the third derivative rate K(s0) forthe restriction v : [0, s0] → (0, ∞) is bounded by:

K(s0) = c3α−1(α + 1)(α + 2)

α2(1 − cα−1α−1s0)α+3

(Remark in Section 4.2.2). Conversely for any K(0) � K0 < ∞, there areunique s0 < S0 so that the equalities K0 = K(s0) hold.

Theorem 1.4 Let us f ix any K0 = K(s0). For any 1 < a ∈ Q and T0 � s0, letu : (0, ∞) × [0, T0) → (0, ∞) be C3 solutions to the dif fusion equations:

us − u2x = ua.

44 T. Kato

Suppose their third derivative rates are bounded by K0. Then for any 0 < ε �(200K0)

−1, u satisfy the asymptotic estimates:(u(x, s)v(s)

)±1

� 1040 aε−2(2x+4s)+1−1a−1 ([u : v]5ε)

aε−2(2x+4s)+4.

Next we treat diffusion equations of the form:

us − u2x − ua − δub = 0, (1 < a < b , 0 < δ << 1)

where we consider the equations of the types:

(a, b) = (2, 3),(1 + α−1, 1 + 2α−1

), (3, 5)

and 0.5 < α < 1 are any rational numbers. Both the right and left hand sideterms touch the Fujita index (= 3), and the middle terms cross it. For exampleit contains the case (a, b) = (2.5, 4).

For 0.5 � α � 1 and c > 0, let us put:

K(s0) = c3α−153(α + 1)(α + 2)

63α2(1 − c′s0)α+3

(0 � s0 <

1

c′

), c′ = 5cα−1

6α, (3)

(a, b) = (1 + α−1, 1 + 2α−1) , δ = με2, μ = α + 1

9α. (4)

For any positive rational numbers μ = pq ∈ Q>0, where p, q ∈ N are relatively

prime, we put cμ ≡ pq ∈ Z>0.Let us compare u with the function:

v(s) = c(1 − c′s)α

.

Let us fix any K0 = K(s0) � K(0).

Theorem 1.5 For any 0 < T0 � s0 and any 0 < ε � (200K0)−1, let u : (0, ∞) ×

[0, T0) → (0, ∞) be C3 solutions to the the dif fusion equations:

us − u2x = ua + δub .

Suppose their third derivative rates are bounded by K0. Then u satisfy theasymptotic estimates:(

u(x, s)v(s)

)±1

� (2Mμ)8 bε−2(2x+4s)+1−1b−1 ([u : v]5ε)

b ε−2(2x+4s)+4

where Mμ = max(2 × 103c2μ, 3 × 104).

These results come from a general procedure of comparison method whichwe will describe below. Our task is to seek for discrete dynamics which induce


desired PDE, but such dynamics are not unique. The analytic conditionsin these results are heavily depend on choices of such discrete dynamics.Particularly of interest for us is to obtain such estimates by use of α + 1derivatives for larger α. If one can find more suitable discrete dynamics, thenone will obtain better asymptotic estimates of solutions. We refer [2], [3] formore concrete cases.

On the other hand in [5] we have constructed some examples of pairs ofPDEs whose particular solutions do not have such uniform bounds mutually,and so which do not arise from the above procedure. Thus our relations onthe uniform bounds for solutions are non trivial among the set of PDEs of 2variables.

1.1.1 Uniform Bounds of Higher Derivative Rates

Functions we consider here are assumed to satisfy uniform boundedness ofhigher derivatives rates. Functions ‘close’ to polynomials will be particularcases.

It follows from the next lemma that there are pairs of Cα+1 functions whichadmit uniformly bounded derivative rates of order α + 1, and still break thesedouble exponential estimates on any large bounded domains:

Lemma 1.6 Let us take any α � 1 and any large C0 >> 0. Then there areconstants cα+1 independent of C0, and pairs of Cα+1[0, 2) functions u, v whosederivative rates of order α + 1 are bounded by cα+1, so that the estimates hold:

v(s)u(s)

{= 1 0 � s � 1,

� C0 2 − (2C0)−1 � s < 2.

Proof Let u : [0, 2) → (0, 2C0] be the linear function by u(s) = C0(2 − x). α +1 derivatives of u vanish for α � 1, and so all higher derivative rates are zero.We construct v : [0, 2) → [0.5C0, 2C0] which satisfies:

v(s) ={

u(s) 0 � s � 1,

0.5C0 1.5 � s � 2.

Let w : [0, 2) → [0.5, 2] be a smooth and non increasing function whichsatisfies w(s) = 2 − s for 0 � s � 1, and ≡ 0.5 for 1.5 � s � 2. Then there areconstants cα+1 so that its derivative rates of order α + 1 are bounded by cα+1.Let us put v : [0, 2) → [0.5C0, 2C0] by v(s) = C0w(s). Then v are the desiredfunctions, since they have the same higher derivative rates as w. This completesthe proof. �

When the domains for (x, s) are unbounded, what we are focusing by thosedouble exponential estimates, is not behavior at infinity for (x, s). In fact itfollows from the assumption of uniform boundedness of higher derivative rates

46 T. Kato

that u will grow at most exponentially. So constrains of the defining equationshave an effective influence on bouneded regions.

1.2 Real Rational Dynamics and Tropical Geometry

A relative (max, +)-function ϕ is a piecewise linear function of the form:

ϕ(x) = max (α1 + a1 x, . . . , αm + amx) − max(β1 + b 1 x, . . . , βl + b l x

)where al x = �n

i=1ailxi, x = (x1, . . . , xn) ∈ Rn, al = (a1

l , . . . , anl ), b ∈ Zn and

αi, βi ∈ R. We say that the multiple integer M ≡ ml is the number of thecomponents of ϕ.

Correspondingly tropical geometry associates the parametrized rationalfunction given by (see [9]):

ft(z) ≡ kt(z)

ht(z)= �m

k=1tαk zak

�lk=1tβk zb k

where za = ni=1zai

i , z = (z1, . . . , zn) ∈ Rn>0. We say that ft above is a relative

elementary function. We say that both terms ht(z) = �lk=1tβk zb k and kt(z) =

�mk=1tαk zak are just elementary functions.These two functions ϕ and ft admit one to one correspondence between

their presentations. Moreover the defining equations are transformed by twosteps, firstly taking conjugates by logt and secondly by letting t → ∞. Noticethat when all ai and bj are zero, then the corresponding ft are t independent.

In some cases the same (max, +) function admits different presentations,while the corresponding rational functions are mutually different. For examplefor ϕ(x) ≡ max(x, x) = x ≡ ψ(x), the corresponding rational functions ft(z) =2z and gt(z) = z are mutually different. We call such a pair of rational functionstropically equivalent.

Let ft : Rn>0 → (0, ∞) be a rarional function, and consider the discrete

dynamics defined by:

zN = ft(zN−n, . . . , zN−1), N � n

with initial values (z0, . . . , zn−1) ∈ Rn>0. One can regard that tropically equiva-

lent rational functions determine the same dynamics at infinity.Let us put:

PN(c) =⎧⎨⎩

cN−n+1 − 1

c − 1c > 1,

N − n + 1 c = 1.

For a relative elementary function ft, let c f � 1 be the Lipschitz constantand M f be the number of the components with respect to the corresponding(max, +)-function.


Our basic analysis on the orbits is given by the following (Corollary 2.8):

Lemma 1.7 Let gt be tropically equivalent to ft, and {zN}N and {wN}N bethe orbits for ft and gt with the initial values z0 = (z0, . . . , zn−1) and w0 =(w0, . . . , wn−1) respectively. Then the estimates hold:

(zN

wN

)±� M4PN(c)

[max

0�i�n−1

(zi

wi

)±1]cN

where c = max(c f , cg) and M = max(M f , Mg).

If the initial values are the same, then uniform estimates hold (Proposi-tion 2.3):

(zN

wN

)±� M2PN(c).

One particular feature is that when the Lipschitz constant is equal to 1, thenthe above inequalites give the exponential estimates, while for c > 1, they aredouble exponential. The former is applied for the estimates of solutions tolinear PDEs.

When one considers evolutional discrete dynamics, a parallel estimates aregiven. An evolutional discrete dynamics is given by flows of the form {zt

N}t,N�0,where t is time parameter. A general equation of evolutional discrete dynamicsis of the form:

zt+1N+1 = f

(zt+1

N−l0, . . . , zt+1

N , ztN−l1

, . . . , ztN+k1

, . . . , zt−dN+kd+1

)

where li, k j � 0, N � max(l0, . . . , ld+1) and t � d, with initial values:

z00 ≡ {

zta

}0�a�max(l0,...,ld+1),t=0,1,...

∪ {zh

N

}0�h�d,N=0,1,...

.

Let us take g tropically equivalent to f , and consider the dynamics {wtn}

defined by g with any initial values w00 . Then we put the initial rates by:

[z0

0 : w00

] ≡ sup0�a�max(l0,...,ld+1),b=0,1,..., or a=0,1,...,0�b�d

{zb

a

wba

,wb

a

zba

}.

Let us put l = max(l0, l1, . . . , ld+1), k = max(k1, . . . , kd+1) and

A(N, t) ≡ (t − d − 1)k + N − l + n − 1

for N � l + 1 and t � d + 1.

48 T. Kato

Proposition 1.8

(1) Let f and g be tropically equivalent. Then any orbits {ztN}N and {wt

N}N forf and g with the initial values z0

0 and w00 respectively, satisfy the estimates:(

ztN

wtN

)±� M4PA(N,t)(c)

[z0

0 : w00

]cA(N,t)

where c = max(c f , cg) and M = max(M f , Mg).(2) Let f, f ′, g, g′ be four relative elementary functions, and assume that they

are all monotone increasing and all tropically equivalent. Let {vtN}N,t and

{utN}N,t be positive sequences so that these satisfy the estimates:

f ′(vt+1

N−l0, . . . , vt−d

N+kd+1

)� vt+1

N+1 � f(vt+1

N−l0, . . . , vt−d

N+kd+1

), (5)

g′(

ut+1N−l0

, . . . , ut−dN+kd+1

)� ut+1

N+1 � g(

ut+1N−l0

, . . . , ut−dN+kd+1

)(6)

for all N, t. Then the ratios satisfy the uniform estimates:(vt

N

utN

)±1

� M8PA(N,t)(c)[u0

0 : v00

]cA(N,t)

.

Here also if the Lipschitz constants c are equal to 1, then the above twoestimates are at most exponential, while for the case c > 1, they are doubleexponential.

Such general form will allow us to treat wider classes of PDE. But forconcrete cases, we use evolutional discrete dynamics only of the forms:

zt+1N+1 =

{f(zt+1

N−1, ztN, zt

N+2

)for quasi linear equations,

f(zt

N, ztN+4, zt−1

N−4, zt−4N−1

)for diffusion equations.

For the former l = 1, k = 2, d = 0, and for the latter l = k = d = 4. So they aregiven by:

A(N, t) ={

2t + N − 1 (N � 2, t � 1),

4t + N − 21 (N � 5, t � 5).

1.3 Rough Approximations by Discrete Dynamics

Let us describe our general procedure for approximating solutions todifferential equations by discrete dynamics, and outline how to verify theoremsin Section 1.1.

Let us consider a Cα+1 function u : (0, ∞) → (0, ∞), and for 1 � |i| � n − 1,take the Taylor expansions:

u(x + iε) = u(x) + iεux + (iε)2

2u2x + · · · + (iε)α

α! uαx + (iε)(α+1)

(α + 1)! u(α+1)x(ξi).


Then for small ε > 0 and N = 0, 1, 2, . . . , let us put

zN ≡ εu(εN) = εu(x),(

N = xε

).

Let f = kh : Rn

>0 → (0, ∞) be a relative elementary function of n variables,where both h and k are elementary, and consider the discrete dynamicsdefined by wN+1 = f (wN−n+1, . . . , wN) with the initial value wi = εu(εi) > 0for 0 � i � n − 1. Our basic idea is to regard that the sequence {wN}N wouldapproximate the orbit {zN}N .

So let us consider the difference and insert the Taylor expansions:

zN+1 − f (zN−n+1, . . . , zN) = εu(x + ε) − f (εu(x − (n − 1)ε), . . . , εu(x)) (7)

= ε

(u + εux + ε2

2u2x + . . .

)− f (ε(u − (n − 1)εux + . . . ), . . . , εu) (8)

= εF1(u) + ε2 F2(ux) + ε3 F3(u, ux) + .. + εm Fm′(u, .., u(α+1)x(ξ)) + ..

h(εu(x − (n − 1)ε), . . . , εu(x))(9)

where Fk are monomials.For any finite subset A ⊂ {1, 2, 3, . . . }, let us divide the expanded sum into

two terms as:

= �i∈Aεsi Fs′i(u, ux, . . . , uαx)

h(εu(x − (n − 1)ε), . . . )+ � j∈Acεs j Fs′

j(u, ux, . . . , u(α+1)x(ξ))

h(εu(x − (n − 1)ε), . . . )(10)

≡ F(ε, u, ux, . . . , uαx) + ε2F1(ε, u, ux, .., u(α+1)x(ξ1), .., u(α+1)x(ξn−1)) (11)

We say that F and F1 are the leading and error terms respectively. Onceone has chosen a relative elementary function f , then the above processdetermines a PDE defined by F, while tropical geometry gives an automatonby a (max, +) function ϕ. So f plays a role of a bridge to connect between PDEand automaton.

Let us define ε variation of F1 by:∥∥F1(ε, u, ux, . . . , uαx, u(α+1)x(ξ1), . . . , u(α+1)x(ξn−1))∥∥

ε(x) ≡ (12)

supμi−x∈I(n,ε)

∣∣F1(ε, u(x − ε), .., uαx(x − ε), u(α+1)x(μ1), .., u(α+1)x(μn−1))∣∣ (13)

where I(n, ε) = [−nε, 0] is the fluctuation interval.Let us say that a Cα+1 function u : (0, ∞) → (0, ∞) is ε controlled, if there

is some constant C > 0 so that ε variation of F1 satisfy the pointwise estimatesfor all x ∈ (0, ∞):

Cu(x) �∥∥F1(ε, u, ux, . . . , uαx, u(α+1)x(ξ1), . . . , u(α+1)x(ξn−1))

∥∥ε(x).

For two functions u, v, we put their initial rates by [u : v]ε ≡supx∈(0,ε]

( u(x)

v(x)

)±1.

50 T. Kato

Now we state the first estimates. Let f and g be relatively elementaryfunctions, and F and G be their corresponding leading terms. Recall thatassociated with f are the Lipschiz constant c f � 1 and the number of thecomponents M f . Let us put c = max(c f , cg) and M = max(M f , Mg).

Proposition 1.9 Let f and g be both relatively elementary and increasingfunctions of n variables, which are mutually tropically equivalent. Let F andG be their leading terms of order at most α � 0, and take positive Cα+1 solutionsu, v : (0, ∞) → (0, ∞) with:

F(ε, u, ux, . . . , uαx) = 0, G(ε, v, vx, . . . , vαx) = 0.


(1

2C , ε0), the estimates hold:(

u(x)

v(x)

)±1

� (2M)8 cε−1x+1−1

c−1 ([u : v]nε)cε−1 x+1

.

One can proceed parallelly for evolutional case. Let f be a relativelyelementary function, and consider the evolutional discrete dynamics definedby the equation zt+1

N+1 = f(zt+1

N−l0, . . . , zt−d

N+kd+1

).

Let us take a Cα+1 function u : (0, ∞) × [0, T0) → (0, ∞), and introduceanother parameters by:

εmu(x, s) = ztN, N = x

ε p, t = s

εq

where p, q � 1 and m � 0 are integers. By the same way as one variable case,one takes the Taylor expansion, and take the difference:

zt+1N+1 − f

(zt+1

N−l0, . . . , zt−d

N+kd+1

)(14)

= εmu(x + ε p, s + εq) − (15)

f(εmu

(x − l0ε

p, s + εq) , . . . , εmu(x + kd+1ε

p, s − dεq)) (16)

= εm F1(u) + εm+p F2(ux) + εm+q F3(us) + ε2m+p F4(u, ux) + . . .

h(εmu(x − l0ε p, s + εq), . . . , εu(x + kd+1ε p, s − dεq))(17)

= F(ε, u, ux, us, uxs, . . . , uαx, uαs) (18)

+ εm+1F1(ε, u(x, s), ux(x, s), . . . , uαx(x, s), (19)

us(x, s), . . . , uαs(x, s), {u(α+1)x(ξij), . . . , u(α+1)s(ξij)}i, j). (20)

By the same way as before one defines the ε variation ||F1||ε(x, s) and ε0

controlledess (Section 3.2.1). Combining this construction with Proposition1.9, one obtains Theorem 1.1.

Our basic process goes as follows. Firstly we choose a PDEF(u, ux, us, . . . ) = 0, and fix scaling parameters. Then find a relative elementaryfunction f which induces F as its leading term. Next take another relative


elementary g which is tropically equivalent to f . Then by use of the samescaling parameters, it induces its leading term G. Finally for two solutions uand v with F(u, ux, us, . . . ) = 0 and G(v, vx, vs, . . . ) = 0 respectively, we seekfor analytic conditions to both u and v which insure ε0 controllednesss. Eventhough choice of f and g are rather flexible, whether one could find somereasonable conditions for solutions depends on choices of these functions.

2 Discrete Dynamics and Tropical Geometry

2.1 Elementary Functions

A relative (max, +)-function ϕ is a piecewise linear function of the form:

ϕ(x) = max (α1 + a1 x, . . . , αm + amx) − max(β1 + b 1 x, . . . , βl + b l x

)where akx = �n

i=1aikxi, x = (x1, . . . , xn) ∈ Rn, ak = (

a1k, . . . , an

k

), b k ∈ Zn and

αk, βk ∈ R.For each relative (max, +) function ϕ as above, we associate a parametrized

rational function by:

ft(z) = �mk=1tαk zak

�lk=1tβk zb k

where zak = ni=1z

aik

i , z = (z1, . . . , zn) ∈ Rn>0 = {(w1, . . . , wn) : wi > 0}.

We say that ft above is a relative elementary function. Notice that anyrelative elementary functions take positive values for z ∈ Rn

>0.We say that the integer:

M ≡ ml

is the number of the components.We say that ft(z) = �m

k=1tαk z jk is an elementary rational function [4]. Thecorresponding (max, +)-function is given by ϕ(x) = max(α1 + j1 x, . . . , αm +jmx), and in this case m is the number of the components.

These two functions ϕ and ft are connected passing through some inter-mediate functions ϕt [7, 9]. Let us describe it shortly below. For t > 1, thereis a family of semirings Rt which are all the real number R as sets. Themultiplications and the additions are respectively given by x ⊕t y = logt(t

x +ty) and x ⊗t y = x + y. As t → ∞ one obtains the equality:

x ⊕∞ y = max(x, y).

By use of Rt as coefficients, one has relative Rt-polynomials:

ϕt(x) = (α1 + a1 x) ⊕t · · · ⊕t (αm + amx) − (β1 + b 1 x

) ⊕t · · · ⊕t(βl + b l x

)

52 T. Kato

The limit is given by the relative (max, +) function above:

limt→∞ ϕt(x) = ϕ(x).

Let us put Logt : Rn>0 → Rn by (z1, . . . , zn) → (logt z1, . . . , logt zn). Then ϕt and

ft satisfy the following relation:

Proposition 2.1 [7, 10] ft ≡ (logt)−1 ◦ ϕt ◦ Logt : Rn

>0 → (0, ∞) is the relativeelementary function ft(z) = �m

k=1tαk zak/�lk=1tβk zb k .

These functions ϕ, ϕt and ft admit one to one correspondence be-tween their presentations. We say that ϕ is the corresponding (max, +)-function to ft. Notice that any relative (max, +) functions of the form ϕ(x) =max(a1 x, . . . , amx) − max(b 1 x, . . . , b l x) correspond to t-independent relativeelementary functions f .

2.2 Discrete Dynamics

Let ft : Rn>0 → (0, ∞) be a relative elementary function, and ϕ be the corre-

sponding (max, +)-function. Let us consider the discrete dynamics defined by:

zN = ft (zN−n, . . . , zN−1) , N � n

with initial values (z0, . . . , zn−1) ∈ Rn>0. These orbits {zN}N admit some asymp-

totic control passing through tropical geometry, which we describe below. Letus compare the orbits {xN}N with {zN}N , which are determined by:

xN = ϕ(xN−n, . . . , xN−1)

with the initial values x0 = logt z0, . . . , xn−1 = logt zn−1. For this, we introducethe intermediate dynamics:

x′N = ϕt

(x′

N−n, . . . , x′N−1

)with the same initial data x′

0 = logt z0, . . . , x′n−1 = logt zn−1.

By Proposition 2.1, two orbits {zN}N and {x′N}N are conjugate each other

as x′N = logt zN for all N = 0, 1, . . . Since limt→∞ ϕt = ϕ holds, one may think

{logt zN}N ‘approximate’ {xN}N in some sense.Let ϕ and ψ be two relative (max, +)-functions with n variables. Then ψ is

equivalent to ϕ, if they are the same as maps, ϕ(x1, . . . , xn) = ψ(x1, . . . , xn) forall (x1, . . . , xn) ∈ Rn (but possibly they can have different presentations).

Definition 2.1 [4] Let ft and gt be two relative elementary functions. gt istropically equivalent to ft, if the corresponding relative (max, +)-functions ϕ

and ψ are equivalent.


Remarks

(1) If the pointwise estimate ϕ � ϕ′ holds, then ψ = max(ϕ, ϕ′) and ϕ areequivalent. Let ft and gt be the corresponding relative elementary func-tions to ϕ and ϕ′ respectively. Then ht ≡ ft + gt is tropically equivalentto ft.

(2) For any relative elementary ft and positive rational numbers 0 < α = nm ∈

Q, α ft is tropically equivalent to ft. In fact let ϕ correspond to ft. Then nft

are tropically equivalent to ft, since nft correspond to max(ϕ, . . . , ϕ) = ϕ

(n times). Similarly 1m ft are also tropically equivalent to ft.

(3) For any tropically equivalent pairs of relative elementary functions ft andgt, the corresponding (max, +)-functions ϕ and ψ have the same Lipschitzconstant c > 0, since they are the same as maps. On the other handthey may have different numbers of the components M and M′ since itdepends on their presentations. For example if ft has M number of thecomponents, then n

m ft has nmM number of the components.(4) For our purposes in this paper, it is enough to treat the case that the

Lipschitz constants c for ϕ is larger or equal to 1, and later on we willassume the bounds c � 1.

2.3 Basic Estimates and Lipschitz Constants

Let ft : Rn>0 → (0, ∞) be a relative elementary function. Take initial val-

ues (z0, . . . , zn−1) ∈ Rn>0, and consider the orbits {zN}∞N=0 defined by zN =

ft(zN−n, . . . , zN−1) for N � n. Let gt be another relative elementary function,and consider its orbit {wN}N with the same initial values wi = zi for 0 � i �n − 1.

In order to estimate their asymptotic rates( zN

wN

)±1 in detail, we use themetric on Rn given by:

d((x0, . . . , xn−1), (y0, . . . , yn−1)) ≡ max0�i�n−1

{|xi − yi|}

(which is of course equivalent to the standard one.)

Lemma 2.2 Let ft = f be t-independent, relative elementary and linear. Thenthe corresponding (max, +)-function ϕ has its Lipschitz constant bounded by 1.

Proof This follows immediately, if one checks the estimates carefully. One canexpress ϕ(x0, . . . , xn−1) = max(α1 + xi1 , . . . , αn−1 + xin−1) − max(0, . . . , 0). Letϕ(x0, . . . , xn−1) = α1 + xi1 � ϕ(y0, . . . , yn−1) = α2 + yi2 . Then the estimateshold:

|ϕ(x0, . . . , xn−1) − ϕ(y0, . . . , yn−1)| = α1 + xi1 − (α2 + yi2) (21)

� α1 + xi1 − (α1 + yi1) = xi1 − yi1 � max0�i�n−1

{|xi − yi|}. (22)

This completes the proof. �

54 T. Kato

In general we have double exponential estimates for( zN

wN

)±1 as below, butin a special case that the Lipschitz constants of the corresponding (max, +)-functions are equal to 1, they can be improved to be just exponential. Thishappens when one considers linear PDE.

Let us put

PN(c) =⎧⎨⎩

cN−n+1 − 1

c − 1c > 1,

(N − n + 1) c = 1.

.

For a relative elementary function ft, let c f be the Lipschitz constant andM f be the number of the components with respect to the corresponding(max, +)-function.

Proposition 2.3 ft and gt are tropically equivalent, if and only if any orbits withthe same intial values satisfy uniformly bounded rates:(

zN

wN

)±1

≡ zN

wN,wN

zN� M2PN(c), (N � n)


For the proof, we use the next lemma.Let ϕ(x) = max

(α1 + a1 x, . . . , αm + amx

) − max(β1 + b 1 x, . . . , βl + b l x

)and ϕt be the corresponding functions to ft. For the same initial valuesx0 = x′

0, . . . , xn−1 = x′n−1, let us denote the orbits by {xN}N and {x′

N}N for ϕ

and ϕt respectively.We will improve Lemma 2.2 in [4] slightly.

Lemma 2.4 Let c � 1 and M be the Lipschitz constant and the number of thecomponents for ϕ respectively. Then the estimates hold:

|xN − x′N| � PN(c) logt M.

Proof One can obtain the following estimates easily [4, Lemma 2.1(1)]:

|ϕ(x0, . . . , xn−1) − ϕt(x0, . . . , xn−1)| � logt M.

Let us denote xN = (xN, . . . , xN+n−1) ∈ Rn. Thus xN+n = ϕ(xN) hold for allN � 0. Similar for x′

N .Firstly one has the estimates |x′

n − xn| � logt M as above.Since ϕ is c-Lipschitz and x1 − x′

1 = (0, . . . , 0, xn − x′n), the estimates:

|xn+1 − x′n+1| = |ϕ(x1) − ϕt(x′

1)| (23)

� |ϕ(x1) − ϕ(x′1)| + |ϕt(x′

1) − ϕ(x′1)| (24)

� c|x1 − x′1| + logt M � (c + 1) logt M (25)


hold. Next we have estimates:

|ϕ(x2) − ϕ(x′2)| � c max

(|xn+1 − x′n+1|, |xn − x′

n|)

� c(c + 1) logt M, (26)

|xn+2 − x′n+2| = |ϕ(x2) − ϕt(x′

2)| (27)

� |ϕ(x2) − ϕ(x′2)| + |ϕ(x′

2) − ϕt(x′2)|

� [c(c + 1) + 1] logt M (28)

The rest is just the repetition of the same process. Now suppose c > 1. Thenby a direct calculation, one obtains the estimates:

|xN − x′N| � cN−n+1 − 1

c − 1logt M.

On the other hand when c = 1, then |xN − x′N| � (N − n + 1) logt M hold. This

completes the proof. �

Proof of Proposition 2.3 The proof is almost the same as Theorem 2.1 in [4],but for convenience we will include only if part.

Let ϕ and ψ be the relative (max, +)-functions corresponding to ft andgt respectively. For the same initial values xi = yi = logt zi, 0 � i � n − 1, letus denote the corresponding orbits by {xN}N and {yN}N . We also put x′

N =logt(zN) and y′

N = logt(wN) respectively. Thus {x′N}N is the orbit for ϕt and

{y′N}N is for ψt.By Lemma 2.4, the estimates:

|xN − x′N|, |yN − y′

N| � PN(c) logt M

hold. Suppose ft and gt are tropically equivalent, and so ϕ and ψ are the sameas maps. Thus xN = yN hold, and so we have the estimates:

logt

(zN

wN

)±� | logt(zN) − logt(wN)| = |x′

N − y′N| � 2PN(c) logt M.

Thus we have the estimates:(zN

wN

)±� max

(zN

wN,wN

zN

)� M2PN(c).


Remarks

(1) In order to determine zN for N � n, one needs to iterate N − n + 1 timesto apply function ft. One can say that ratios between N − n + 1 timesiterations of ft and gt are at most uniformly double exponential rates.

(2) Such double exponential estimates are optimal between tropically equiv-alent functions. Let us consider two dynamics for l, k � 1:

zN = f (zN−1) = zlN−1, wN = g(wN−1) = 2wk

N−1.

56 T. Kato

If l = k holds, then f and g are tropically equivalent. Let z0 = w0 be initialvalues. Then a direct calculation gives:

zN = zlN

0 , wN = 2kN−1k−1 wkN

0 = 2kN−1k−1 zkN

0 .

Thus if l = k, then the equality:(wN

zN

)±1

= 2± lN−1l−1

holds, which satisfies the uniformly double exponential bound.On the other hand if k > l, then

wN

zN= 2

kN−1k−1 zkN−lN

0

which heavily depends on the initial values.

Lemma 2.5 Let ft and gt be relative elementary and assume that both aremonotone increasing. Let {vN}N be a positive sequence so that the estimates:

gt(vN−n, . . . , vN−1) � vN � ft(vN−n, . . . , vN−1), N � n

hold. Let {zN}N and {wN}N be two dynamics def ined by zN = ft(zN−n, . . . ,

zN−1) and wN = gt(wN−n, . . . , wN−1) with the same initial value zi = wi = vi for0 � i � n − 1 respectively. Then the estimates hold:

wN � vN � zN (N = 0, 1, . . . )

Proof We proceed by induction. For N = n, the estimates follows by thehypothesis. Suppose the estimates wN � vN � zN hold for N � N0 − 1. Thenthe conclusion for N0 follows from two estimates:

wN0 = gt(wN0−n, . . . , wN0−1) � gt(vN0−n, . . . , vN0−1), (29)

zN0 = ft(zN0−n, . . . , zN0−1) � ft(vN0−n, . . . , vN0−1) (30)

and the assumption gt(vN0−n, . . . , vN0−1) � vN0 � ft(vN0−n, . . . , vN0−1).This completes the proof. �

Corollary 2.6 Let ft and gt be tropically equivalent, and assume the conditionsin Lemma 2.4 are satisf ied. Then the estimates hold:(

zN

vN

)±1

,

(wN

vN

)±1

� M2PN(c)


Proof By Lemma 2.5, the estimates wN � vN � zN hold for all N = 0, 1, . . .

On the other hand by Proposition 2.3, the uniform bounds zNwN

� M2PN(c) hold.Then the conclusions follow from the estimates zN

vN� zN

wN� M2PN(c) and vN

wN�

zNwN

� M2PN(c). This completes the proof. �


For example gt = 1m ft are the cases for m � 1, when ft is monotone

increasing.

2.3.1 Dependence on Initial Values

Let ft : Rn>0 → (0, ∞) be a relative elementary function. Let us take two initial

values:

z0 = (z0, . . . , zn−1), w0 = (w0, . . . , wn−1) ∈ Rn>0

and consider the corresponding orbits {zN}∞N=0 and {wN}∞N=0 defined by:

zN = ft(zN−n, . . . , zN−1), wN = ft(wN−n, . . . , wN−1), (N � n)

respectively. Let ϕ and ϕt be the functions corresponding to ft.Here we have more elaborate estimates:

Proposition 2.7 Let ft and the orbits {zN}∞N=0, {wN}∞N=0 be as above with initialvalues z0 and w0. Then they satisfy uniformly bounded rates:

(zN

wN

)±� M2PN(c)

[max

0�i�n−1

(zi

wi

)±1]cN

(N � n)

where c and M are the Lipschitz constant and the number of the components forϕ respectively.

Proof The idea of the proof is parallel to Proposition 2.3.Let us put x′

N = logt(zN) and y′N = logt(wN) respectively. Thus {x′

N}N is theorbit for ϕt with the initial value x′

i = logt zi for 0 � i � n − 1, and similar for{y′

N}N .Let {xN}N be another orbit for ϕ with the same initial value xi = logt zi for

0 � i � n − 1, and similar for {yN}N .Let c � 1 be the Lipschitz constant for ϕ. Let us estimate |xN − yN| for N �

n. Since xn = ϕ(x0, . . . , xn−1) and yn = ϕ(y0, . . . , yn−1), the estimate:

|xn − yn| = |ϕ(x0, . . . , xn−1) − ϕ(y0, . . . , yn−1)| � c max0�i�n−1

|xi − yi|

hold. Let us iterate the same estimates:

|xn+1 − yn+1| = |ϕ(x1, . . . , xn) − ϕ(y1, . . . , yn)| (31)

� c max1�i�n

|xi − yi| � c2 max0�i�n−1

|xi − yi|. (32)

The same process gives us the estimates:

|xN − yN| � cN−n+1 max0�i�n−1

|xi − yi| = cN−n+1 max0�i�n−1

logt

(zi

wi

)±1

.

On the other hand by Lemma 2.4, the estimates:

|xN − x′N|, |yN − y′

N| � PN(c) logt M

58 T. Kato

hold, where M is the number of the components for ϕ. So combining with theseestimates, we have the followings:

max

(logt

zN

wN, logt

wN

zN

)= | logt(zN) − logt(wN)| = |x′

N − y′N| (33)

� |xN − x′N| + |yN − y′

N| + |xN − yN| (34)

� 2PN(c) logt M + cN max0�i�n−1

logt

(zi

wi

)±1

(35)

= logt

⎧⎨⎩M2PN(c)

[max

0�i�n−1

(zi

wi

)±1]cN

⎫⎬⎭ . (36)

Thus one obtains the estimates:

(zN

wN

)±1

� M2PN(c) max0�i�n−1

(zi

wi

)±cN

.


Now let gt and ft be two relatively elementary functions, and denote thecorresponding pairs of the functions by (ϕ, ϕt) and (ψ, ψt) respectively. Let(c f , M f ) and (cg, Mg) be the Lipschitz constants and the numbers of thecomponents for ϕ and ψ respectively.

Corollary 2.8 Let gt be tropically equivalent to ft, and {zN}N and {wN}N bethe orbits for ft and gt with the initial values z0 = (z0, . . . , zn−1) and w0 =(w0, . . . , wn−1) respectively. Then the estimates hold:

(zN

wN

)±� M4PN(c)

[max

0�i�n−1

(zi

wi

)±1]cN


Proof Let {z′N}N be the orbit for ft with the initial value w0 = (w0, . . . , wn−1).

By Proposition 2.7, one obtains the estimates:

(zN

z′N

)±1

� M2PN(c)

[max

0�i�n−1

(zi

wi

)±1]cN

.

On the other hand by Proposition 2.3, one has another estimates:

(z′

N

wN

)±1

� M2PN(c).


By multiplying both sides, one obtains the desired estimates:

(zN

wN

)±1

=(

zN

z′N

)±1 (z′

N

wN

)±1

� M2PN(c)M2PN(c)

[max

0�i�n−1

(zi

wi

)±1]cN

(37)

= M4PN(c)

[max

0�i�n−1

(zi

wi

)±1]cN

. (38)


Now we induce the main estimates:

Theorem 2.9 Let us take four relative elementary functions, ft, f ′t , gt, g′

t. As-sume that they are all monotone increasing and all tropically equivalent. Let{vN}N and {uN}N be positive sequences which satisfy the estimates:

f ′t (vN−n, . . . , vN−1) � vN � ft(vN−n, . . . , vN−1), (39)

g′t(uN−n, . . . , uN−1) � uN � gt(uN−n, . . . , uN−1). (40)

for all N � n. Then the ratios satisfy the uniform estimates:

(vN

uN

)±1

� M8PN(c)

[max

0�i�n−1

(ui

vi

)±1]cN

where c = max(c f , c f ′ , cg, cg′) and M = max(M f , M f ′, Mg, Mg′).

Proof Let us consider two orbits {zN}N and {z′N}N defined by zN = ft(zN−n,

. . . , zN−1) and z′N = f ′

t

(z′

N−n, . . . , z′N−1

)with the the same initial value zi =

z′i = vi for 0 � i � n − 1 respectively. Similarly by use of gt and g′

t, one hasorbits for {wN}N and {w′

N}N with the initial value wi = w′i = ui for 0 � i � n − 1

respectively.Then by Corollary 2.6, one has the estimates:

(zN

vN

)±1

,

(wN

uN

)±1

� M2PN(c).

On the other hand by Corollary 2.8, the estimates hold:

(zN

wN

)±1

� M4PN(c)

[max

0�i�n−1

(zi

wi

)±1]cN

= M4PN(c)

[max

0�i�n−1

(vi

ui

)±1]cN

.

60 T. Kato

Thus from these two, one obtains the desired uniform estimates:(vN

uN

)±1

=(

vN

zN

)±1 (zN

wN

)±1 (wN

uN

)±1

(41)

� M2PN(c)M4PN(c)

[max

0�i�n−1

(vi

ui

)±1]cN

M2PN(c) (42)

= M8PN(c)

[max

0�i�n−1

(vi

ui

)±1]cN

. (43)


2.4 Evolutional Dynamics

For simplicity of the notation, later on we will omit to denote the parameter tfor ft and just write f for any relative elementary functions.

Let f be a relative elementary function. A general equation of evolutionaldiscrete dynamics is of the form:

zt+1N+1 = f

(zt+1

N−l0, . . . , zt+1

N , ztN−l1

, . . . , ztN+k1

, zt−1N−l2

, . . . , zt−1N+k2

, . . . , zt−dN+kd+1

)where li, k j � 0, N � max(l0, . . . , ld+1) and t � d, with initial values:

z00 ≡ {

zta

}0�a�max(l0,...,ld+1),t=0,1,...

∪ {zh

N

}0�h�d,N=0,1,...

.

As before one puts the Lipschitz constant and the number of the compo-nents by c f and M f for the corresponding (max, +)-function to f .

Let us put l = max(l0, l1, . . . , ld+1), k = max(k1, . . . , kd+1) and

A(N, t) ≡ (t − d − 1)k + N − l + n − 1

for N � l + 1 and t � d + 1.Let us take g tropically equivalent to f , and consider the dynamics {wt

n}defined by g with any initial value w0

0 .Now we put the initial rates by:

[z0

0 : w00

] ≡ sup0�a�max(l0,...,ld+1),b=0,1,..., or a=0,1,...,0�b�d

{zb

a

wba

,wb

a

zba

}.

Proposition 2.10

(1) Let f and g be tropically equivalent. Then any orbits {ztN}N and {wt

N}N forf and g with the initial values z0

0 and w00 respectively, satisfy the estimates:(

ztN

wtN

)±� M4PA(N,t)(c)

[z0

0 : w00

]cA(N,t)



(2) Let f, f ′, g, g′ be four relative elementary functions, and assume that theyare all monotone increasing and all tropically equivalent. Let {vt

N}N,t and{ut

N}N,t be positive sequences so that these satisfy the estimates:

f ′(vt+1

N−l0, . . . , vt−d

N+kd+1

)� vt+1

N+1 � f(vt+1

N−l0, . . . , vt−d

N+kd+1

), (44)

g′(

ut+1N−l0

, . . . , ut−dN+kd+1

)� ut+1

N+1 � g(

ut+1N−l0

, . . . , ut−dN+kd+1

)(45)

for N � l and t � d. Then the ratios satisfy the uniform estimates:(vt

N

utN

)±1

� M8PA(N,t)(c)[u0

0 : v00

]cA(N,t)

for N � l + 1 and t � d + 1.

Proof Let us check that in order to determine zd+tl+N , one has to iterate at

most (t − 1)k + N times to apply f for N, t � 1. Then the conclusions followfrom Corollary 2.8 and Theorem 2.9 (see remark (1) below the proof ofProposition 2.3).

Let us denote by α(N, t) the number of compositions of f in order todetermine zt

N . It is an increasing function on both variables. We show theestimates α(l + N, d + t) � (t − 1)k + N.

Let �0 = {(a, b) ∈ {0, 1, . . . , k + l} × {0, 1, . . . , d} ∪ {0, . . . , l} × {d + 1}} bethe finite set. This is a basic building block in the sense that for N, t � 1, zt+d

N+l

is determined if one knows zt−1+bN−1+a for (a, b) ∈ �0.

We proceed by induction on t. α(l + N, d + 1) � N clearly follows.Suppose the conclusion follows for t � t0, and so α(N + l, d + t0) �

(t0 − 1)k + N hold. Then α(l + 1, d + t0 + 1) = α(l + k, d + t0) + 1 � (t0 −1)k + k + 1 = t0k + 1 hold. Next α(l + 2, d + t0 + 1) = max(α(l + 1, d + t0 +1), α(l + k + 1, d + t0)) + 1 � t0k + 2. By use of the estimates α(N + l, d + t0 +1) � max(α(N − 1 + l, d + t0 + 1), α(N − 1 + l + k, d + t0)) + 1, one can ob-tain the bounds α(N + l, d + t0 + 1) � t0k + N.


3 Asymptotic Comparisons

3.1 Formal Taylor Expansion and ODE

Let us consider a Cα+1 function u : (0, ∞) → (0, ∞). Below we proceed to ap-proximate u very roughly by discrete dynamics defined by relative elementaryfunctions of n variables. For 1 � |i| � n − 1, let us take the Taylor expansionsaround x ∈ (0, ∞):

u(x + iε) = u(x) + iεux + (iε)2

2u2x + · · · + (iε)α

α! uαx + (iε)(α+1)

(α + 1)! u(α+1)x(ξi)

62 T. Kato

for small |ε| << 1, where: {x � ξi � x + iε, i � 0

x + iε � ξi � x i < 0

(for our applications, we will choose α � 2 later).Let f = k

h : Rn>0 → (0, ∞) be a relative elementary function, where h and k

are both elementary. Later on we will assume positivity:

h(0) > 0.

Let us consider the discrete dynamics defined by zN+1 = f (zN−n+1, . . . , zN).We put the f luctuation intervals by:

I(n, ε) = [−nε, 0].For N = 0, 1, 2, . . . , let us put change of variables:

zN ≡ εu(εN) = εu(x),(

N = xε

).

Let us consider the difference:

zN+1 − f (zN−n+1, . . . , zN) = εu(x + ε) − f (εu(x − (n − 1)ε), . . . , εu(x))

and insert the Taylor expansions:

= ε

(u + εux + ε2

2u2x + . . .

)(46)

− f(

ε

(u − (n − 1)εux + (n − 1)2ε2

2u2x + . . .

), . . . , εu

). (47)

By reordering the expansions with respect to the exponents of ε, there arerational numbers a0, a1, · · · ∈ Q so that the equality holds:

εu(x + ε) − f (εu(x − (n − 1)ε), . . . , εu(x)) (48)

= εa0u + ε2a1ux + ε3a2uux + .. + εα+1asuαx + εα+2as+1u(α+1)x(ξ) + ..

h(εu(x − (n − 1)ε), . . . , εu(x))(49)

≡ εF1(u) + ε2 F2(ux) + ε3 F3(u, ux) + .. + εm Fm′(u, .., u(α+1)x(ξ)) + ..

h(εu(x − (n − 1)ε), . . . , εu(x))(50)

where Fk are monomials.Let us choose finite subsets A ⊂ {1, 2, 3, . . . }, and divide the expanded sum

into two terms as:

= �i∈Aεsi Fs′i(u, ux, . . . , uαx)

h(εu(x − (n − 1)ε), . . . )+ � j∈Acεs j Fs′

j(u, ux, . . . , u(α+1)x(ξ))

h(εu(x − (n − 1)ε), ..)(51)

≡ F(ε, u, ux, . . . , uαx) + ε2F1(ε, u, ux, .., u(α+1)x(ξ1), .., u(α+1)x(ξn−1)). (52)


We always choose A so that two conditions are satisfied; (1) F do not containu(l+1)x(ξ), and (2) 1 ∈ A, i.e. F1 is included in F.

In all the concrete cases later, we choose relative elementary functions andA so that the corresponding F1 vanish.

Now fix ε > 0, and suppose u obeys the equation:

F(ε, u, ux, . . . , uαx) = 0.

Then the difference satisfies the equality:

εu(x + ε) − f (εu(x − (n − 1)ε), . . . , εu(x)) = ε2F1(ε, u, ux, . . . ).

We say that F is the leading term, and F1 error one for u respectively.

Remark Conversely when one starts from ODE F(ε, u, ux, uαx) = 0, there willbe several choices of relative elementary functions f and A with the leadingterm F. Various choices of f will assign different error terms F1, which reflectestimates of solutions F(ε, u, ux, . . . , uαx) = 0. So ‘better’ choice of f will giveus ‘better’ estimates of large scale analysis of such solutions.

Let us define ε variation of F1 by∥∥F1(ε, u, ux, . . . , uαx, u(α+1)x(ξ1), . . . , u(α+1)x(ξn−1))∥∥

ε(x) (53)

≡ supμi−x∈I(n,ε)

∣∣F1(ε, u(x − ε), .., uαx(x − ε), u(α+1)x(μ1), .., u(α+1)x(μn−1))∣∣ (54)

where I(n, ε) is the fluctuation interval.Let us say that a Cα+1 function u : (0, ∞) → (0, ∞) is ε0 controlled, if there

is some constant C > 0 so that the ε0 variation of F1 satisfy the pointwiseestimates for all x ∈ (0, ∞):

Cu(x) �∣∣∣∣F1(ε, u, ux, . . . , uαx, u(α+1)x(ξ1), . . . , u(α+1)x(ξn−1))

∣∣∣∣ε0

(x)

3.1.1 Comparison Theorem for ODE

Let us take another relatively elementary function g = de which is tropically

equivalent to f . Let v : (0, ∞) → (0, ∞) be another Cα+1 function. By re-placing f by g and choosing another subsets B ⊂ {1, 2, 3, . . . } in Section 3.1,one has its leading and error terms G and G1 respectively. Then we have theequalities:

εv(x + ε) − g(εv(x − (n − 1)ε), . . . , εv(x)) (55)

= �i∈Bεsi Gs′i(v, vx, . . . , vαx)

e(εv(x − (n − 1)ε), . . . )+ � j∈Bcεs j Gs′

j(v, vx, . . . , v(α+1)x(ξ′))

e(εv(x − (n − 1)ε), . . . )(56)

≡ G(ε, v, . . . , vαx) + ε2G1(ε, v, . . . , vαx, v(α+1)x(ξ′1), . . . , v(α+1)x(ξ

′n−1)). (57)

64 T. Kato

Let us fix a small ε > 0, and take two positive solutions u, v : (0, ∞) →(0, ∞) to the equations:

F(ε, u, ux, . . . ) = 0, G(ε, v, vx, . . . ) = 0.

Now we compare their ratios:(u(x)

v(x)

)±1

={

u(x)

v(x),v(x)

u(x)

}.

For this we introduce the initial rates:

[u : v]ε ≡ supx∈(0,ε]

(u(x)

v(x)

)±1

.

Recall that associated with f are the Lipschitz constant c f � 1 and the num-ber of the components M f . Let us put c = max(c f , cg) and M = max(M f , Mg).

Theorem 3.1 Let f and g be both relatively elementary and increasing functionsof n variables, which are mutually tropically equivalent. Let F and G be theirleading terms of order at most α � 0, and take positive Cα+1 solutions u, v :(0, ∞) → (0, ∞) to the equations:

F(ε, u, ux, . . . , uαx) = 0, G(ε, v, vx, . . . , vαx) = 0.


(1

2C , ε0), the estimates hold:(

u(x)

v(x)

)±1

� (2M)8 cε−1x+1−1

c−1 [u : v]cε−1 x+1

nε .

Proof Let f and g be both n variables, and (F, F1) and (G, G1) be pairs ofleading and error terms respectively.

Let us choose 0 < ε � min(

12C , ε0

). By the assumption, the pointwise esti-

mates hold:

Cu(x + ε) �∣∣F1(ε, u(x), ux(x), . . . , uαx(x), u(α+1)x(ξ1), . . . , u(α+1)x(ξn−1))

∣∣ .In particular the estimates ε2|F1| � 1

2C ε|F1| � 12εu(x + ε) hold.

Let us consider the equalities:

εu(x + ε) − f (εu(x − (n − 1)ε), . . . , εu(x)) (58)

= F(ε, u, . . . , uαx) + ε2F1(ε, u, . . . , uαx, u(α+1)x(ξ1), . . . , u(α+1)x(ξn−1)) (59)

= ε2F1(ε, u, ux, . . . , uαx, u(α+1)x(ξ1), . . . , u(α+1)x(ξn−1)) (60)

since u obeys the equation F(ε, u, ux, . . . ) = 0.Then combining with the above inequality, one obtains the estimates:

1

2f (εu(x − (n − 1)ε), . . . , εu(x)) � εu(x + ε) (61)

� 2 f (εu(x − (n − 1)ε), . . . , εu(x)). (62)


By the same way one obtains the estimates by replacing f by g:

1

2g(εv(x − (n − 1)ε), . . . , εv(x)) � εv(x + ε) (63)

� 2g(εv(x − (n − 1)ε), . . . , εv(x)). (64)

f , 12 f and 2 f are tropically equivalent, and 1

2 f, 2 f, 12 g, 2g are all so by the

assumption. Notice that the number of the components for 12 f and 2 f are both

2M f .Thus the estimates hold by Theorem 2.9:(

u(Nε)

v(Nε)

)±1

� (2M)8PN(c) sup0�i�n−1

(u(εi)v(εi)

)±cN

� (2M)8PN(c)([u : v](n−1)ε)cN

.

For any 0 � μ � ε, let us apply the above estimates for the translationsu(x + μ) and v(x + μ). Then one obtains the estimates:(

u(Nε + μ)

v(Nε + μ)

)±1

� (2M)8PN(c)[u : v]cN

nε (65)

= (2M)8 cN−n+1−1c−1 [u : v]cN

nε

� (2M)8 cε−1(Nε+μ)−n+2−1

c−1 [u : v]cε−1(Nε+μ)+1

nε (66)

since PN(c) = cN−n+1−1c−1 .

Such Nε + μ takes all the points x ∈ (0, ∞), and so the estimates hold:(u(x)

v(x)

)±1

� (2M)8 cε−1 x−n+2−1

c−1 [u : v]cε−1 x+1

nε � (2M)8 cε−1x+1−1

c−1 [u : v]cε−1 x+1

nε .


Example Let us consider a simple equation:

F(u, ux) = ux + u2 = 0.

It has solutions u(x) = a1+ax with the initial values u(0) = a > 0. Let us put

zN = εu(x) with x = Nε and take the Taylor expansion εu(x + ε) = εu(x) +ε2ux(x) + ε3

2 u2x(ξ). We choose the relative elementary function f (x) = x(1 +x)−1 and calculate the difference:

u(x + ε) − f (εu(x)) (67)

= ε2 ux + u2

1 + εu(x) + ε3

12 u2x(ξ) + u(x)ux(x) + ε

2 u(x)u2x(ξ)

1 + εu(x)(68)

Thus u is ε0 controlled, since 1 + εu(x) � 1, and the estimates |u2x(ξ)|, |uux(x)|,|u(x)u2x(ξ)| � Cu(x) hold uniformly in x ∈ (0, ∞) for some C = C(a, ε0) � 0and |x − ξ | � ε.

The corresponding (max, +) function to f is given by VN+1 = VN −max(0, VN). Notice the equality VN − max(0, VN) = VN − max(0, VN, VN).

66 T. Kato

The tropical inverse for the latter is given by g(y) = y(1 + 2y)−1. By choosingthe same scaling parameter, one obtains the leading term G(v, vx) = vx + 2v2.It has solutions v(x) = a′

2a′x+1 , and the ratio is in fact uniformly bounded:

(u(x)

v(x)

)±=

(a(2a′x + 1)

a′(ax + 1)

)±1

≤ 2( a

a′)±1

� 2[u : v]ε .

3.2 Evolutional Dynamics

Here we treat partial differential equations. The process of Section 3.2 is quiteparallel to Section 3.1 by introducing time parameter.

A general equation of evolutional discrete dynamics is of the form:

zt+1N+1 = f

(zt+1

N−l0, . . . , zt+1

N , ztN−l1

, . . . , ztN+k1

, zt−1N−l2

, . . . , zt−1N+k2

, . . . , zt−dN+kd+1

)where li, k j � 0, N � l ≡ max(l0, . . . , ld+1) and t � d, with initial values:

z00 ≡ {

zta

}0�a�l,t=0,1,...

∪ {zh

N

}0�h�d,N=0,1,...

.

Now let us consider a Cα+1 function u : (0, ∞) × [0, T0) → (0, ∞), andintroduce another parameters by

N = xε p

, t = sεq

, εmu(x, s) = ztN

where ε > 0 is a small constant, and p, q � 1, m � 0 are integers. Then we takethe Taylor expansions:

u(x + iε p, s + jεq) = u + iε pux + jεqus + (iε p)2

2u2x + ( jεq)2

2u2s (69)

+ jεqiε puxs + · · · + (iε p)α

α! uαx + ( jεq)α

α! uαs (70)

+ (iε p)(α+1)

(α + 1)! u(α+1)x(ξij) + · · · + ( jεq)(α+1)

(α + 1)! u(α+1)s(ξij) (71)

≡ u + iε pux + jεqus + (iε p)2

2u2x + ( jεq)2

2u2s

+ jεqiε puxs (72)

+ · · · + (iε p)α

α! uαx + ( jεq)α

α! uαs

+ �a(iε p)a( jεq)α+1−a

(α + 1)! ua(ξij) (73)

where a = (yi1 , . . . , yiα+1), y j = x or s, and |(x, s) − ξij| � |(iε p, jεq)|.


Let f = kh : Rn

>0 → (0, ∞) be a relative elementary function, and considerthe difference as in Section 3.1:

zt+1N+1 − f

(zt+1

N−l0, . . . , zt+1

N , ztN−l1

, . . . , ztN+k1

, . . . , zt−dN+kd+1

)(74)

= εmu(x + ε p, s + εq) (75)

− f(εmu

(x − l0ε

p, s + εq) , . . . , εmu(x + kd+1ε

p, s − dεq)) . (76)

By reordering the expansions with respect to the exponents of ε, there arerational numbers a0, a1, · · · ∈ Q so that the above difference is equal to thefollowing:

εm a0u + ε pa1ux + εqa2us + εm+pa3uux + .. + (iε p)h( jεq)α+1−hahuh(ξij) + ..

h(εmu(x − l0ε p, s + εq), . . . , εmu(x + kd+1ε p, s − dεq))

(77)

≡ εm F1(u) + εm+p F2(ux) + εm+q F3(us) + ε2m+p F4(u, ux) + . . .

h(εmu(x − l0ε p, s + εq), . . . , εmu(x + kd+1ε p, s − dεq))(78)

where Fk are monomials.Let us choose finite subsets A ⊂ {1, 2, 3, . . . }, and divide the expanded sum

into two terms as:

εmu(x + ε p, s + εq) (79)

− f(εmu

(x − l0ε

p, s + εq) , . . . , εmu(x + kd+1ε

p, s − dεq)) (80)

= εm F1(u) + εm+p F2(ux) + εm+q F3(us) + ε2m+p F4(u, ux) + . . .


= �i∈Aεsi Fs′i(u, ux, us, . . . , uαs) + � j∈Acεs j Fs′

j(u, ux, . . . , ua(ξij))


≡ F(ε, u, ux, us, . . . , uαs) + εm+1F1(ε, u, ux, . . . , {ua(ξij)}a,i, j). (83)

As in Section 3.1, we always choose A so that F do not contain ua(ξ) and 1 ∈ A.We call F as the leading term and F1 the error term respectively.

3.2.1 ε-Controlledness

Now we return to the starting point. Let f be a relative elementary function,

and consider the discrete dynamics zt+1N+1 = f

(zt+1

N−l0, . . . , zt+1

N , ztN−l1

, . . . , ,

zt−dN+kd+1

). After one chooses integers p, q, m for change of variables, one

determines the leading and error terms F and F1 respectively.For (a, b) ∈ Z2, let L(a, b) = {(ta, tb) : t ∈ [0, 1]} ⊂ R2 be the segment.

Then for the set:

D ≡{(1, 1), (−l0, 1), . . . , (0, 1), (−l1, 0), . . . , (k1, 0),

(−l2, −1), . . . , (k2, −1), . . . , (−ld+1, −d), . . . , (kd+1, −d)}

68 T. Kato

we put the f luctuation domain as:

D(ε, p, q) = {(L

(ε pa, εqb

) : (a, b) ∈ D} ⊂ R2.

For example D = {(1, 1), (2, 0), (−1, 1)} for zt+1N+1 = f

(zt

N, ztN+2, zt+1

N−1

).

Let us regard F1 as a function on the variables (x, s, {ξij}i, j). Then we defineits ε variation:∥∥F1

∥∥ε(x, s) ≡ sup

ξij−(x−ε p,s−εq)∈D(ε,p,q)

∣∣F1(ε, u

(x − ε p, s − εq) , (84)

ux(x − ε p, s − εq) , us

(x − ε p, s − εq) , . . . , uαs

(x − ε p, s − εq) , {ua(ξij)})

∣∣ .(85)

Let u : (0, ∞) × [0, T0) → (0, ∞) be a Cα+1 function.

Definition 3.1 u is ε0 controlled bounded by C, if ε0 variation of F1 satisfies thepointwise estimates:

Cu(x, s) �∣∣∣∣F1

∣∣∣∣ε0

(x, s)

for all (x, s) ∈ (0, ∞) × [0, T0).

Higher Derivative Rates Let u : (0, ∞) × [0, T0) → (0, ∞) be a Cα+1 functionand f be a relative elementary function. let us consider the expansions of thedifferences in Section 3.2:

εmu(x + ε p, s + εq) (86)

− f(εmu

(x − l0ε

p, s + εq) , . . . , εmu(x + kd+1ε

p, s − dεq)) (87)

= F (ε, u, ux, us, . . . , uαs) + εm+1F1(ε, u, ux, . . . , {ua(ξij)}a,i, j

). (88)

F has order at most α, while F1 may contain derivatives of u smaller than α + 1in general.

Let us say that the error term F1 is admissible, if it is of the form:

F1 = �a∈Ac caεsa Ha

(εmu

(x − l0ε

p, s + εq) , . . .)

ua(ξij)

where (1) |a| = α + 1 and (2) ||Ha(x1, x2, . . . )||C0 � 1 for any x1, x2 · · · � 0.For this case we put the error constants by:

CF1 ≡ �a∈Ac |ca| ∈ Q>0.

The error constants are determined by the coefficients of rational functions fand of the Taylor expansions. Our applications later are all admissible cases.

Let us introduce variation of order α + 1 of u by:

||u||α+1(x, s) = max∂i= ∂x, ∂s

{sup

ξ−(x−ε p,s−εq)∈D(ε,p,q)

∣∣∣∣ ∂α+1u∂1 . . . ∂α+1

∣∣∣∣ (ξ)

}.

Let us say that u satisfies uniform ε variation, if there is a constant C so thatit satisfies the estimates:

Cu(x, s) � ||u||α+1(x, s)


for all (x, s) ∈ (0, ∞) × [0, T0).We put the variation constant by:

V(u) ≡ sup(x,s)∈(0,∞)×[0,T0)

||u||α+1(x, s)u(x, s)

.

Lemma 3.2 Suppose F1 is admissible, and u satisf ies uniform ε variationbounded by C. Then u is ε controlled bounded by CCF1 .

Proof By admissibility, the estimates hold:

||F1||ε(x, s) � �a∈Ac |ca|εsa∣∣Ha

(εmu

(x − l0ε

p, s + εq) , . . .)∣∣ ||u||α+1(x, s)

� �a∈Ac |ca|||u||α+1(x, s) � CF1 ||u||α+1(x, s) � CF1 Cu(x, s).

�

Let u : (0, ∞) × [0, T0) → (0, ∞) be a Cα+1 function. Here we considerclasses of functions which satisfy uniform rates between higher derivatives andlowest values.

Let us introduce the derivative constants of α + 1, which is given by:

||u||α+1 = max∂i= ∂x, ∂s

{∥∥∥∥ ∂α+1u∂1 . . . ∂α+1

∥∥∥∥C0((0,∞)×[0,T0))

}.

Suppose u satisfies two conditions:(1) ||u||α+1 < ∞ is finite and (2) c = inf(x,s)∈(0,∞)×[0,T0) u(x, s) > 0 is positive.

Then we say that the ratio:

K(u) ≡ ||u||α+1

c

is the derivative rates of order α + 1. In general the estimates hold:

V(u) � K(u).

Now we state the following which requires more practical conditions:

Lemma 3.3 Suppose F1 is admissible, and the derivative rates of order α + 1 isof f inite, K(u) < ∞. Then for any ε > 0, u is ε controlled bounded by CF1 K(u).

Proof By the conditions, the error term F1 satisfies the estimates:

||F1||ε(x, s) � CF1 ||u||α+1(x, s) � CF1 ||u||α+1

= CF1 K(u)c � CF1 K(u)u(x, s).

So u is ε-controlled bounded by CF1 K(u) for any ε > 0.This completes the proof. �

70 T. Kato

3.2.2 Comparison Theorem

Let g be tropically equivalent to f , and choose the same scaling parameters.By replacing f by g above, one obtains another leading and error terms G andG1 with the equalities:

εmv(x + ε p, s + εq) (89)

− g(εmv

(x − l0ε

p, s + εq) , . . . , εmv(x + kd+1ε

p, s − dεq)) (90)

= G(ε, v, vx, vs, . . . , vαs) + εm+1G1(ε, v, vx, vs, . . . , vαs, {va(ηij)}a,i, j). (91)

Let us fix a small ε > 0, and take two positive solutions u, v : (0, ∞) ×[0, T0) → (0, ∞) to the corresponding PDEs:


In order to estimate their ratios( u(x,s)

v(x,s)

)±1, we introduce the initial rates:

[u : v]ε ≡ sup(x,s)∈(0,∞)×[0,εq]∪(0,ε p]×[0,T0)

(u(x, s)v(x, s)

)±1

.

Recall the Lipschitz constant c f and the number of the components M f forf . Let us put c = max(c f , cg), M = max(M f , Mg), k = max(k1, . . . , kd+1) andL = max(l, d) for l = max(l0, l1, . . . , ld+1).

Corollary 3.4 Let f and g be both relatively elementary and increasing functionsof n variables, which are mutually tropically equivalent. Let F and G be theirleading terms of order at most α � 0, and take positive Cα+1 solutions u, v :(0, ∞) × [0, T0) → (0, ∞) to the equations:



(1

2C , ε0)

and D = max(p, q), the estimates hold:

(u(x, s)v(x, s)

)±1

� (2M)8 cε−D(x+ks)+1−1

c−1 [u : v]cε−D(x+ks)+n

(L+1)ε .

Proof Recall A(N, t) = (t − d − 1)k + N − l + n − 1 for N � l + 1 and t �d + 1. Let us take 0 < ε � min

(1

2C , ε0). Combining with Proposition 2.10, the

parallel argument to the proof of Theorem 3.1 gives the estimates:

(u (Nε p + μ, tεq + χ)

v (Nε p + μ, tεq + χ)

)±1

� (2M)8PA(N,t)(c)[u : v]cA(N,t)

(L+1)ε


for any 0 � μ � ε p and 0 � χ � εq. Then we have the estimates:

A(N, t) = (t − d − 1)k + N − l + n − 1 (92)

� ε−qk(tεq + χ

) − dk + ε−p (Nε p + μ

) − l + n (93)

� ε−qk(tεq + χ

) + ε−p (Nε p + μ

) + n (94)

� ε−α[k

(tεq + χ

) + (Nε p + μ

)] + n (95)

where α = max(p, q). Then:

(2M)8PA(N,t)(c)[u : v]cA(N,t)

(L+1)ε (96)

≤ (2M)8 cε−D[k(tεq+χ)+(Nε p+μ)]+1−1

c−1 [u : v]cε−D[k(tεq+χ)+(Nε p+μ)]+n

(L+1)ε . (97)

Now combing with these estimates, one obtains the desired estimates:(u(x, s)v(x, s)

)±1

� (2M)8 cε−D(x+ks)+1−1

c−1 [u : v]cε−D(x+ks)+n

(L+1)ε .


Example Let b > a � 1 be positive integers, and consider linear PDEsF(vx, vs) = avx + bvs = 0. For increasing and relative elementary functions fwith its leading term F, let us consider the discrete dynamics:

zt+1N+1 = f

(zt

N, ztN+1

) = 1

b

(azt

N + (b − a)ztN+1

).

Let v : (0, ∞) × [0, ∞) → (0, ∞) be C2 functions, and take the Taylor expan-sions up to order 2. We choose the scaling parameters by N = x

ε, t = s

εand

ztN = v(x, s), and insert the Taylor expansions:

v(x + ε, s + ε) − f (v(x, s), v(x + ε, s)) (98)

= ε

b(avx + bvs) + ε2

2

((v2x + v2s + 2vxs)(η1) − b − a

bv2x(η2)

). (99)

f correspond to Vt+1N+1 = max

(Vt

N, . . . , VtN, Vt

N+1, . . . , VtN+1

) − max(0, . . . , 0),where their terms iterate a, b − a and b times respecively. Clearly this showsthat f are all tropically equivalent indpendently of b > a � 1.

By Lemma 2.2, the Lipschitz constants c f = 1 are all equal to one. For thenumbers of the components, M f = b 2 hold. D = 1, L = 0, n = 2 and k = 1.For any positive integers a, b , a′, b ′, let us take two solutions u(x, s) and v(x, s)satisfying the equations aux + bus = 0 and a′vx + b ′vs = 0 respecively. Onemay assume b � b ′. Suppose both are ε0 controlled bounded by C. Then byCorollary 3.4, for any 0 < ε � min

(1

2C , ε0), the exponential estimates must

hold: (u(x, s)v(x, s)

)±1

� (2b 2)8(ε−1(x+s)+1)[u : v]ε .

72 T. Kato

Below we apply the general procedure of the previous sections to non linearpartial differential equations. We treat two PDEs, where one is the quasi linearequations of order 1, and the other is diffusion equations. Given PDE, thenour procedure is to find ‘good’ relative elementary functions f . We have torequire them to be increasing. Any elementary polynimials are increasing. Oneof applicable form of f is:

f (z1, z2, . . . ) = z1(α + P(z1, z2, . . . ))

1 + z1+ Q(z1, . . . )

where both P and Q are elementary polynomials and 0 � α � 1.One may weaken the required properties, if both the range and the domain

for discrete dynamics are within the regions of monotone increasing for thesefunctions.

4 Applications

4.1 Quasi Linear Equations

Here we introduce a cancelation method of non linear terms and use itto compare solutions between the following equations. Let us consider theequations of the form:

vs + εvvx − 1

2v2 = 0, 2us + εu(us + ux) = 0

where ε > 0 are small constants. These two types of the equations differfrom each other, in that for the right hand side, each monomial containsdifferentials of u, and so in particular any constants are solutions. Notice thatv(x, s) = c

1−0.5cs are degenerate solutions on (0, ∞) × [0, 2c ) for c > 0.

We choose the second variation:

||u||2(x, s) = supξ−(x−ε,s−ε)∈D(ε,1,1)

{∣∣∣∣∂2u∂x2

∣∣∣∣ (ξ),

∣∣∣∣∂2u∂s2

∣∣∣∣ (ξ),

∣∣∣∣ ∂2u∂x∂s

∣∣∣∣ (ξ)

}

and put the variation constant:

V(u) ≡ sup(x,s)∈(0,∞)×[0,T0)

||u||2(x, s)u(x, s)

.

Let us fix any positive constant V0 > 0.

Theorem 4.1 For any 0 < ε � 0.1V−10 , let v, u : (0, ∞) × [0, T0) → (0, ∞) be

C2 solutions to the quasi linear equations:

vs + εvvx − 1

2v2 = 0, 2us + εu(us + ux) = 0.


Suppose their variation constants V(u), V(v) are bounded by V0. Then theysatisfy the asymptotic estimates for all (x, s) ∈ (0, ∞) × [0, T0):(

u(x, s)v(x, s)

)±1

� 402ε−1(x+2s)+4([u : v]2ε)

2ε−1(x+2s)+3.

In particular when u(x, s) ≡ R > 0 is constant, then the estimates hold:

R(40)−2ε−1(x+2s)+4([v : R]2ε)

−2ε−1(x+2s)+3(100)

� v(x, s) � R(40)2ε−1(x+2s)+4([v : R]2ε)

2ε−1(x+2s)+3. (101)

4.1.1 Induction of the Equations

Let us consider the dynamics:

zt+1N+1 = f

(zt

N, ztN+2, zt+1

N−1

) ≡ ztN+2

2+ zt

N

(1 + 2zt+1

N−1

)2(1 + zt

N

) .

f is an increasing function. The corresponding (max, +)-function is given bymax

(Vt

N+2, VtN+2 + Vt

N, VtN, Vt

N + Vt+1N−1, Vt

N + Vt+1N−1

) − max(0, 0, Vt

N, VtN

).

The number of the components is M = 5 × 4 = 20, and its Lipschitz constantis equal to 2.

We choose the scaling parameters by:

εv(x, s) = ztN, N = x

ε, t = s

ε

where we take a small ε > 0 so that the estimate ε � 0.1V−10 holds.

Let v : (0, ∞) × [0, T0) → (0, ∞) be a C2 function, and take the Taylorexpansions up to order 2:

v(x + iε, s + jε) (102)

= v + iεvx + jεvs + ε2

(i2

2v2x + j2

2v2s + ijvxs

)(ξij) (103)

≡ v + iεvx + jεvs + ε2 D2v(ξij). (104)

Let us insert the formal Taylor expansions:

εv(x + ε, s + ε) − f (εv(x, s), εv(x + 2ε, s), εv(x − ε, s + ε)) (105)

= 1

2(1 + εv)

[ε2(2vs + 2εvvx − v2) − 2(εv)ε2 D2v(ξ−11)}

](106)

+ ε2

(D2v(ξ11) − 1

2D2v(ξ20)

)(107)

where the leading term is given by:

F = ε2 2vs + 2εvvx − v2

2(1 + εv).

74 T. Kato

The error term is admissible, and let us calculate the error constant CF1 . Noticethe estimates |D2v(ξij)| �

( i2+ j2

2 + |ij|)||v||2(x, s), where ||v||2(x, s) is the secondvariation. Then the error term satisfies the estimates:

||F1||ε(x, s) � 2(εv)ε2|D2v(ξ−11)|2(1 + εv)

+ ε2

(∣∣D2v(ξ11)∣∣ +

∣∣∣∣1

2D2v(ξ20)

∣∣∣∣)

� 5ε2||v||2(x, s).

In particular the error constant is given by:

CF1 = 5.

4.1.2 Deformation and Cancelation

Let us introduce a cancelation method below. Let us consider the discretedynamics:

wt+1N+1 = g

(wt

N, wtN+2, w

t+1N−1

) ≡ wtN+2

2+ wt

N + wtNwt+1

N−1

2(1 + wtN)

.

g is also an increasing function and is tropically equivalent to f . The numberof the components is 16, and the corresponding (max, +)-function has itsLipschitz constant 2.

Let u : (0, ∞) × [0, T0) → (0, ∞) be a C2 function, and choose the samescaling parameters, εu(x, s) = zt

N , N = xε

and t = sε. Then let us insert the

Taylor expansions of u up to order 2 into the difference as before. Then thedirect calculation shows that unlike to the previous case, u2 term is eliminated,and the result is given by:

εu(x + ε, s + ε) − g(εu(x, s), εu(x + 2ε, s), εu(x − ε, s + ε)) (108)

= 1

2(1 + εu)

[ε2(2us + εuus + εuux) − εuε2 D2u(η−11)

](109)

+ ε2

(D2u(η11) − 1

2D2u(η20)

)(110)

where the leading term is given by:

G = ε2 2us + εuus + εuux

2(1 + εu).

In this deformation also, the error term is admissible, and satisfies theestimates:

||G1||ε(x, s) �εuε2

∣∣D2u(η−11)∣∣

2(1 + εu)+ ε2

(∣∣D2u(η11)∣∣ +

∣∣∣∣1

2D2u(η20)

∣∣∣∣)

� 4ε2||u||2(x, s).

So the error constant is give by CG1 = 4.


Proof of Theorem 4.1 Let u, v : (0, ∞) × [0, T0) → (0, ∞) be C2 functionswhich satisfy the equations vs + εvvx − 1

2v2 = 0 and 2us + εu(us + ux) = 0.Suppose they have bounded variation constants V(u), V(v) � V0. Then by

applying Corollary 3.4 and Lemma 3.2, one obtains the asymptotic estimates:

(u(x, s)v(x, s)

)±1

� (2M)8 cε−D(x+ks)+1−1

c−1 ([u : v](L+1)ε)cε−D(x+ks)+n

for any 0 < ε � (2CV0)−1, where in this case D = max(p, q) = 1, C = 5, L = 1,

M = 20, c = 2, k = 2 and n = 3. Thus for any 0 < ε � 0.1V−10 , the estimates:

(u(x, s)v(x, s)

)±1

� (40)2ε−1(x+2s)+4([u : v]2ε)

2ε−1(x+2s)+3

hold. This completes the proof. �

4.2 Diffusion Equations

Here we introduce a linear deformation of elementary functions, and use it tocompare between solutions to different diffusion equations.

Let F be a relative elementary and increasing function of one variable. Herewe consider the diffusion equations of the type:

us = u2x + F(u).

We take the third variation:

||u||3(x, s) = supξ−(x−ε2,s−ε)∈D(ε,2,1)

{∣∣∣∣∂3u∂x3

∣∣∣∣ (ξ),

∣∣∣∣∂3u∂s3

∣∣∣∣ (ξ),

∣∣∣∣ ∂3u∂x2∂s

∣∣∣∣ (ξ),

∣∣∣∣ ∂3u∂x∂s2

∣∣∣∣ (ξ)

}

and put the variation constant:

V(u) ≡ sup(x,s)∈(0,∞)×[0,T0)

||u||3(x, s)u(x, s)

.

4.2.1 Linear Deformations

Let F be relative elementary and increasing, or zero. We consider the discretedynamics of the form:

zt+1N+1 = f

(zt−4

N−1, zt−1N−4, zt

N, ztN+4

) = αzt−1N−4 + βzt−4

N−1 + γ ztN + δzt

N+4 + F(zt

N

)where α, β, γ, δ > 0 are all positive rational numbers.

We choose the scaling parameters by

ztN = εlu(x, s), N = x

εm, t = s

ε2m, (l � 0, m � 1).

76 T. Kato

For a C3 function u : (0, ∞) × [0, T0) → (0, ∞), let us take the Taylor expan-sions as before:

u(x + iεm, s + jε2m) = u + iεmux + jε2mus (111)

+ i2 ε2m

2u2x + j2

ε4m

2u2s + ijε3muxs + ε3m D3

εu(ξij). (112)

Firstly we consider the differences:

zt+1N+1 −

(p4

zt−1N−4 + 1 − p

4zt−4

N−1

)for 0 < p < 1. It is immediate to see that this does not contain uxs term. Letus determine p ∈ Q so that it also contains no u2s term. In fact for p = 4

5 , thedifference is:

3

4εlu + 7

5εl+2mus + 37

20εl+mux − 9

8εl+2mu2x + εl+3m Higher terms

where Higher terms consisted of linear combinations of three derivatives.Next we eliminate ux term by adding δzt

N+4 for δ = 3780 , and then finally we

eliminate u terms by adding γ ztN for γ = 23

80 :

zt+1N+1 −

(1

5zt−1

N−4 + 1

20zt−4

N−1 + 37

80zt

N+4 + 23

80zt

N

)(113)

= εl+2m(

7

5us − 193

40u2x

)+ εl+3m Higher terms . (114)

Next if we choose constants as below, then one induces the following:

wt+1N+1 − g

(wt−4

N−1, wt−1N−4, w

tN, wt

N+4

)(115)

≡ wt+1N+1 −

(1

24wt−1

N−4 + 5

24wt−4

N−1 + 1

128wt

N+4 + 95

128wt

N

)(116)

= εl+m(

15

8εmvs + 43

32vx − 19

16ε3mv2s

)+ εl+3m Higher terms. (117)

g and f above are mutually tropically equivalent. If one exchanges the role ofvariables and regards x as the time parameter, then the first term of the righthand side equation gives the advection–dif fusion equation.

Proposition 4.2 Let us f ix V0 > 0 and choose any 0 < ε � (200V0)−1. Let u, v :

(0, ∞) × [0, ∞) → (0, ∞) be C3 solutions to the linear equations:

7

5us − 193

40u2x = 0,

15

8εvs + 43

32vx − 19

16ε3v2s = 0.

Suppose their variation constants satisfy the bounds V(u), V(v) � V0. Thenthey satisfy the exponential asymptotic estimates for all (x, s) ∈ (0, ∞) × [0, ∞):(

u(x, s)v(x, s)

)±1

� 1048(ε−2(x+4s)+1)[u : v]5ε .


Proof Let us consider two linear functions:

f(zt−4

N−1, zt−1N−4, zt

N, ztN+4

) = 1

5zt−1

N−4 + 1

20zt−4

N−1 + 37

80zt

N+4 + 23

80zt

N, (118)

g(wt−4

N−1, wt−1N−4, w

tN, wt

N+4

) = 1

24wt−1

N−4 + 5

24wt−4

N−1 + 1

128wt

N+4 + 95

128wt

N (119)

Let us choose m = 1. Then the estimates in Corollary 3.4 and Lemma 3.2give the following for 0 < ε � (2CV0)

−1:(u(x, s)v(x, s)

)±1

� (2M)8 cε−D(x+ks)+1−1

c−1 ([u : v](L+1)ε)cε−D(x+ks)+n

.

For the corresponding (max, +)-functions, their Lipschitz constants are bothc = 1, and the numbers of the components are bounded roughly by M � 106

2 .For both cases, the error terms are the Higher terms above, consisted by thelinear combinations of the three derivatives. So the error constants are roughlybounded by 1

6 × 8 × 43 � 102. k = 4, D = max(p, q) = 2 and L = max(l, d) =4. So in this case for any 0 < ε � (200V0)

−1, the estimates:(u(x, s)v(x, s)

)±1

� 1048(ε−2(x+4s)+1)[u : v]5ε

hold. This completes the proof. �4.2.2 Non Deforming

Let us consider the non linear diffusion equations:

us = u2x + ua, 1 < a ∈ Q.

In order to estimate its asymptotics, one considers v : (0, ∞) × [0, T0) →(0, ∞), which is a C3 solution to the equation vs = va. For the initial valuec > 0, this is easily solved as:

v(s) = c

(1 − ca−1(a − 1)s)(a−1)−1 .

The blowing up time is S0 = 1ca−1(a−1)

. Its three derivative is given by d3v(s)ds3 =

c1+3α−1(α+1)(α+2)

α2(1−cα−1α−1s)α+3

, where α = (a − 1)−1. Thus for 0 � s � s0 < S0, the variationconstant V is bounded by:

V(s0) = c3α−1(α + 1)(α + 2)

α2(1 − cα−1α−1s0)3

.

Conversely for any V(0) � V0 < ∞, there are unique s0 < S0 so that theequality V0 = V(s0) holds.

Theorem 4.3 Let us f ix any V0 = V(s0). For any 1 < a ∈ Q and T0 � s0, letu : (0, ∞) × [0, T0) → (0, ∞) be C3 solutions to the dif fusion equations:

us − u2x = ua.

78 T. Kato

Suppose their variation constants V(u) are bounded by V0. Then for any 0 <

ε � (200V0)−1, u satisf ies the asymptotic estimates:

(u(x, s)v(s)

)±1

� 1040 aε−2(2x+4s)+1−1a−1 ([u : v]5ε)

aε−2(2x+4s)+4.

Proof In Section 4.2.1, let us choose the rescaling parameters m = 1 and l ∈ Q

so that the equality l + 2m = la holds. In order to induce the above non lineardiffusion equations from discrete dynamics, we add non linear term.

Let u : (0, ∞) × [0, T0) → (0, ∞) be a C3 function, and consider the discretedynamics:

zt+1N+1 − f

(zt−4

N−1, zt−1N−4, zt

N, ztN+4

)(120)

= zt+1N+1 −

(1

5zt−1

N−4 + 1

20zt−4

N−1 + 37

80zt

N+4 + 23

80zt

N + 7

5

(zt

N

)a)

(121)

= εla(

7

5us − 193

40u2x − 7

5ua

)+ εl+3 Higher terms. (122)

For the corresponding (max, +)-function to f , the Lipschitz constant is a>1,and the number of the components are bounded roughly by 1

2 105. The errorconstant is again roughly bounded by 102, k = 4, D = 2 and L = 4.

Suppose u satisfies the equation 75 us − 193

40 u2x − 75 ua = 0 which admits

bounded variation constants V(u) � V. Then by Corollary 3.4 and Lemma 3.2,one finds the asymptotic estimates for 0 < ε � (200V)−1:

(u(x, s)v(s)

)±1

� 1040 aε−2(x+4s)+1−1a−1 ([u : v]5ε)

aε−2(x+4s)+4.

Let us change the variable x as u(x, s) = u(px, s), where 12 � p =√

40193 × 7

5 � 1, and put the variation constants of u by V(u). Notice that if us −u2x − ua = 0 holds, then u satisfies the equation 7

5 us − 19340 u2x − 7

5 ua = 0. Theirvariation constants satisfy the estimates V(u) � V(u). Thus the asymptoticestimates hold for 0 < ε � (200V0)

−1:

(u(x, s)v(s)

)±1

� 1040 aε−2(2x+4s)+1−1a−1 ([u : v]5ε)

aε−2(2x+4s)+4.


Remark Notice that the third derivative rates for v : [0, s0] → (0, ∞) are givenby:

K(s0) = c3α−1(α + 1)(α + 2)

α2(1 − cα−1α−1s0)α+3

.


4.2.3 Inhomogeneous Non Linear Equations

The above method does not work for diffusion equations with inhomogeneousnon linear terms. In order to treat such cases, we use tropical deformations forrelative elementary functions. Here we treat diffusion equations of the form:

us − u2x − ua − δub = 0, (1 < a < b , 0 < δ << 1).

Here we cover the equations of the types:

(a, b) = (1 + α−1, 1 + 2α−1), 0.5 � α � 1.

Let μ = pq ∈ Q>0 be positive rational numbers, where p, q ∈ N are relatively

prime numbers. We put cμ ≡ pq ∈ Z>0 and call them as the number of thecomponents for μ.

For α ∈ Q and c > 0, let us put:

V(s0) = c3α−153(α + 1)(α + 2)

63α2(1 − c′s0)3, c′ = 5cα−1

6α, (123)

(a, b) = (1 + α−1, 1 + 2α−1), δ = με2, μ = α + 1

9α(124)

for 0 � s0 < (c′)−1. Let us compare u with the function:

v(s) = c(1 − c′s)α

.

Let us fix any V0 = V(s0) � V(0).

Theorem 4.4 For any 0 < T0 � s0 and any 0 < ε � (200V0)−1, let u : (0, ∞) ×

[0, T0) → (0, ∞) be C3 solutions to the the dif fusion equations:

us − u2x = ua + δub .

Suppose their variation constants V(u) are bounded by V0. Then u satisf iesthe asymptotic estimates:

(u(x, s)v(s)

)±1

� (2Mμ)8 bε−2(2x+4s)+1−1b−1 ([u : v]5ε)

b ε−2(2x+4s)+4

where Mμ = max(2 × 103c2

μ, 3 × 104).

80 T. Kato

Proof Firstly let us consider the tropical deformation:

wt+1N+1 − g

(wt−4

N−1, wt−1N−4, w

tN, wt

N+4

)(125)

= wt+1N+1 −

(1

25

(wt−1

N−4 + wt−4N−1

) + 1

25wt

N+4 + 22

25wt

N + (wt

N

)a + μ(wt

N

)b)

(126)

= εl(

ε2m(

6

5vs + 4

25ε2mv2s

)+ 26

25εmvx + 17

25ε3mvxs − 4

25ε2mv2x

)(127)

−((

wtN

)a + μ(wt

N

)b)

+ εl+3m Higher terms (128)

=[εl+2m

(6

5vs + 4

25ε2mv2s

)− εalva − μεblvb

](129)

+ εl(

26

25εmvx + 17

25ε3mvxs − 4

25ε2mv2x

)+ εl+3m Higher terms. (130)

Since μ � 13 hold, the number of the components for g is bounded by 75 ×

25c2μ � 2 × 103c2

μ. The corresponding Lipschitz constant is b . D = max(p, q) =2m and L = 4. The error constants are bounded by 3×8

25×6 × 43 � 11.

Sublemma 4.5 For (a, b) = (1 + α−1, 1 + 2α−1), 0.5 � α � 1, one can choosel ∈ Q and m = 1 so that both the equalities l + 2m = al and (b − a)l = 2m hold.

Proof By the condition, m = b−a2 l must hold. By inserting into the first con-

dition, one obtains the equality 1 + (b − a) = a, which certainly hold for theabove pairs (a, b). This completes the proof. �

If one chooses l ∈ Q and m = 1 as above, then the equality holds:

εl+2

(6

5vs + 4

25ε2v2s

)− εalva − μεblvb = εl+2

(6

5vs + 4

25ε2v2s − va − με2vb

).

Sublemma 4.6 Moreover let us put μ = α+19α

. Then for any c > 0, v(s) = c(1−c′s)α(

c′ = 5cα−1

6α

)satisfy the equations:

6

5vs + 4

25ε2v2s − va − με2vb = 0.

This can be checked by direct calculations. Notice it satisfies the equationεl+2m

(65vs + 4

25ε2mv2s) − εalva − μεblvb + εl

(a1ε

mvx + a2ε3mvxs − a3ε

2mv2x)=0.


Proof of Theorem Three derivative of v is given by d3v(s)ds3 = c1+3α−1

53(α+1)(α+2)

63α2(1−c′s)α+3 .Thus for 0 � s � s0 < S0 = (c′)−1, the variation constants are bounded by:

V(s0) = c3α−153(α + 1)(α + 2)

63α2(1 − c′s0)3.

As before for any V(0) � V0 < ∞, there are unique s0 < S0 so that the equalityV0 = V(s0) holds.

For 0 < T0 � s0, let u : (0, ∞) × [0, T0) → (0, ∞) be C3 functions, and con-sider the discrete dynamics:

zt+1N+1 − f

(zt−4

N−1, zt−1N−4, zt

N, ztN+4

)(131)

= zt+1N+1 −

(1

5zt−1

N−4 + 1

20zt−4

N−1 + 37

80zt

N+4 + 23

80zt

N + 7

5(zt

N)a + 7

5(zt

N)b)

(132)

= εla(

7

5us − 193

40u2x − 7

5ua − 7

5ε2ub

)+ εl+3 Higher terms . (133)

f and g above are mutually tropically equivalent.For the corresponding (max, +)-function to f , the Lipschitz constant is b >

1, and the number of the components are bounded roughly by 3 × 104. Theerror constant is roughly bounded by 102. k = 4, D = 2 and L = 4.

Suppose u satisfies the equation 75 us − 193

40 u2x − 75 ua − 7

5δub = 0, and admitsbounded variation constants V(u) � V.

Now let us put Mμ = max(2 × 103c2μ, 3 × 104). Then by Corollary 3.4 and

Lemma 3.2, one finds the asymptotic estimates for any 0 < ε � (200V)−1:(u(x, s)v(s)

)±1

� (2Mμ)8 bε−2(x+4s)+1−1b−1 ([u : v]5ε)

b ε−2(x+4s)+4.

The rest of the proof is the same as Theorem 4.3 just by changing the

variable x as u(x, s) ≡ u(px, s), where p =√

40193 × 7

5 . Then for the variation

constants V(u) � V0 for u and for any 0 < ε � (200V0)−1, the estimates hold:(

u(x, s)v(s)

)±1

� (2Mμ)8 bε−2(2x+4s)+1−1b−1 ([u : v]5ε)

b ε−2(2x+4s)+4.


References

1. Fujita, H.: On the blowing up of solutions of the Cauchy problem for ut = �u + u1+α . J. Fac.Sci. Univ. Tokyo (I) 13, 109–124 (1966)

2. Hirota, R.: Nonlinear partial differential equations I, III. J. Phys. Soc. Jpn. 43, 2074–2086(1977)

3. Kato, T.: Pattern formation from projectively dynamical systems and iterations by families ofmaps. To appear in the Proceedings of the 1st MSJ-SI, Probabilistic Approach to Geometry

4. Kato, T.: Deformations of real rational dynamics in tropical geometry. Geom. Funct. Anal.19(3), 883–901 (2009)

82 T. Kato

5. Kato, T., Tsujimoto, S.: A rough analytic relation on partial differential equations.arXiv:1004.4981v1 math.MG (2010)

6. Lee, T.-Y., Ni, W.-M.: Global existence, large time behaviour and life span of solutions of asemilinear parabolic Cauchy problem. Trans. Am. Math. Soc. 333, 365–378 (1992)

7. Litvinov, G., Maslov, V.: The correspondence principle for idempotent calculus and somecomputer applications. In: Gunawardena, J. (ed.) Idempotency, pp. 420–443. CambridgeUniversity Press, Cambridge (1998)

8. Logan, J.: An Introduction to Nonlinear Partial Differential Equations. Wiley, New York(2008)

9. Mikhalkin, G.: Amoebas and tropical geoemtry. In: Donaldson, S., Eliashberg, Y., Gromov, M.(eds.) Different Faces of Geometry. Kluwer, Norwell (2004)

10. Viro, O.: Dequantization of real algebraic geometry on logarithmic paper. In: Proc. of theEuropean Congress of Math. (2000)

http://arXiv.org/abs/math/1004.4981v1 math.MG


Translating Solitons of Mean Curvature Flowof Noncompact Submanifolds

Guanghan Li · Daping Tian · Chuanxi Wu

Received: 31 March 2010 / Accepted: 5 January 2011 / Published online: 13 January 2011© Springer Science+Business Media B.V. 2011

Abstract We prove the existence and asymptotic behavior of rotationallysymmetric solitons of mean curvature flow for noncompact submanifolds inEuclidean and Minkowski spaces, which generalizes part of the correspondingresults for hypersurfaces of Jian.

Keywords Translating soliton · Mean curvature flow ·Spacelike submanifold

Mathematics Subject Classifications (2010) 53C21 · 53C40 · 58C44 · 35K55

Research partially supported by NSFC (No.10971055), Project of Hubei ProvincialDepartment of Education (No.T200901) and Funds for Disciplines Leaders of Wuhan.

G. Li (B)School of Mathematics and Computer Science, and Key Laboratoryof Applied Mathematics of Hubei Province, Hubei University,Wuhan, 430062, People’s Republic of Chinae-mail: [email protected]

D. TianSchool of Mathematics and Computer Science, Hubei University,Wuhan, 430062, People’s Republic of Chinae-mail: [email protected]

C. WuInstitute of Mathematics, Hubei University, Wuhan, 430062,People’s Republic of Chinae-mail: [email protected]

84 G. Li et al.

1 Introduction

Let M be a smooth manifold of dimension m ≥ 2, and F : M → M be a smoothsubmanifold immersed into an (m + n)-dimensional Riemannian or pseudo-Riemannian manifold M. The mean curvature flow is a smooth family of mapsFt = F(·, t) evolving according to

⎧⎨

⎩

ddt

F(x, t) = H(x, t), x ∈ M,

F(·, 0) = F,

(1)

where H is the mean curvature vector of Mt = Ft(M).The mean curvature flow of hypersurfaces (i.e. (1) with n = 1) in a

Riemannian or Pseudo-Riemannian manifold has been extensively studied inthe last two decades (e.g. [8]). Recently, mean curvature flow of submanifoldswith higher co-dimensions has been paid much attention to, and lots of workhas been done in this field, see [2, 15, 17, 20, 22, 23, 25] for examples. Ifthe ambient space is flat, we can define the so-called translating solitons (see[7, 14, 19]) of the mean curvature flow (1).

Definition 1 A submanifold M in a flat space M is called a translating solitonif there exists a constant vector V in M such that H + T = V on M, where Tis the component of V tangent to M, and H is the mean curvature vector ofM in M. An equivalent equation is H = V⊥, where V⊥ is the projection ofV in M to the normal bundle of M. The 1-parameter family of submanifoldsMt defined by Mt = M + tV for t ∈ R is then a solution to the mean curvatureflow (1). At this time, we say that the solution Mt to the flow (1) moves byvertical translation, and V is called a translating vector.

Singularities in mean curvature flow are generally locally modeled onsoliton solutions such as mean convex hypersurfaces in Euclidean spaces [9]and Lagrangians in complex space forms [7, 19]. Therefore translating solitonshave also been extensively studied for mean curvature flow of hypersurfaces aswell as high co-dimensional submanifolds, see [7, 9–12, 14, 19] for examples. In[6, 11, 12], Jian and so on studied the rotationally symmetric translating solitonsof the mean curvature flow of hypersurfaces (or spacelike hypersurfaces) in aEuclidean space (or Minkowski space). In this paper, we consider a class ofrotationally symmetric translating solitons of mean curvature flow of high co-dimensional noncompact submanifolds and spacelike submanifolds, and provesome similar theorems as that of hypersurfaces, which generalizes part of theresults in [11, 12].

In Section 2, we first recall some fundamental facts on mean curvature flowof high co-dimensional submanifolds, and then state our main results. Thesolutions to a system of perturbed ODEs are studied in Section 3. The maintheorems’ proofs are given in Section 4.

Translating Solitons of Mean Curvature Flow 85

2 Preliminaries and Main Results

In this section, we shall recall some fundamental facts on submanifolds ofMinkowski spaces and Euclidean spaces. The Minkowski space Rm+n

n withindex n is the linear space Rm+n endowed with the Lorentz metric

ds2 =m∑

i=1

(dxi)2 −

m+n∑

α=m+1

(dxα)2,

where {xi, xα} are the standard Euclidean coordinates in Rm+n. Here we agreewith the obvious indices range: 1 � i, j, · · · � m and m + 1 � α, β, · · · � m + n.From now on, we assume the ambient space M is either the Minkowski spaceRm+n

n , or the Euclidean space Rm+n.First we consider the Minkowski space case. An m-spacelike submanifold

in Rm+nn is an m dimensional Riemannian manifold, with everywhere timelike

normal frame fields. Locally, such a submanifold can be expressed as a graphof a vector-valued function f = f (x1, · · · , xm) : Rm → Rn satisfying the space-like condition |Df (x)| < 1 for all x ∈ Rm, where D is the ordinary derivativein Euclidean spaces. We remark that spacelike hypersurfaces in Minkowskispaces have been the subject of investigation in general relativity theory (seefor example [1]). The Bernstein type property of maximal spacelike graphsimmersed into a curved pseudo-Riemannian manifold is studied in [16].

Assume F : Rm → Rm+nn is an m-dimensional noncompact spacelike sub-

manifold immersed in Rm+nn . The mean curvature flow is the evolution of a

family immersions Ft = F(·, t) : Rm → Rm+nn satisfying the evolution equation

ddt

F(x, t) = H(x, t), F(x, 0) = F0(x) = F(x), (2)

where H(x, t) is the mean curvature vector of the spacelike submanifoldMt = F(Rm, t) at (x, t). We note that along the mean curvature flow, thespacelike condition is preserved as long as the solutions exist [25]. For compactcase, this is proved in [15]. In terms of the local coordinates (x1, · · · , xm) onRm, the induced metric on Mt is given by gij = 〈 ∂ F

∂xi ,∂ F∂x j 〉, which is positive

definite because Mt is spacelike. Denote by ∇ and the induced Levi-Civitaconnection and Laplacian on Mt, respectively, we have

H = F = gij∇i∇ jF = gij ∂2 F∂xi∂x j

− gij�kij

∂ F∂xk

,

where � is the Christoffel symbol of Mt and gij is the inverse of gij. Thereforethe mean curvature flow is the solution

F = {F A (

x1, · · · , xm, t), A = 1, · · · , m + n

},

to the following system of parabolic equations

ddt

F =⎛

⎝m∑

i, j=1

gij ∂2 F∂xi∂x j

⎞

⎠

⊥

, (3)

86 G. Li et al.

where the notion ⊥ denotes the projection of a vector in Rm+nn onto the

orthogonal complement of the tangent space.Denote the projection π : Rm+n

n → Rm. Assume F is the solution to themean curvature flow (2). If each Mt = F(Rm, t) can be written as a space-likegraph over Rm, then rt = (π ◦ Ft)

−1 is a family of diffeomorphism of Rm. Using(3), by a similar discussion as in [24], we see that F = F(r(x, t), t) is a repara-metrization of F and satisfies the following system of parabolic equations

ddt

F = gij ∂2 F∂xi∂x j

. (4)

According to the description above, there exists a family of functions f α,

m + 1 � α � m + n on Rm such that F(x1, · · · , xm) = (x1, · · · , xm, f m+1, · · · ,

f m+n), and the mean curvature flow (2) satisfies the parabolic equations⎧⎪⎨

⎪⎩

df α

dt= gij ∂2 f α

∂xi∂x j, α = m + 1, · · · , m + n,

f (·, 0) = f0, f = (f m+1, · · · , f m+n

)(5)

for f0 ∈ C∞(Rm), where gij is the inverse of the induced metric gij, and in termsof f , the induced metric is given by

gij = δij −∑

α

∂ f α

∂xi

∂ f α

∂x j. (6)

Conversely, if (5) is satisfied for some f , then F = I × f satisfies (4).Moreover, there exists a family of diffeomorphisms rt of Rm such that F(x, t) =F(r(x, t), t) satisfies the mean curvature flow (2). For more details, see [24]. Weremark that the tangent part of the mean curvature flow (4) does not effect thegeometry of the corresponding evolved spacelike submanifolds.

We now assume the solution Mt to the mean curvature flow (2) moves byvertical translation, i.e., by Definition 1, there exists a constant vector V ofRm+n

n such that

Mt = M + tV,

for any t ∈ R. When n = 1, i.e. for spacelike hypersurfaces, translating solitonscan be regarded as a natural way of foliating spacetimes by almost null likehypersurfaces, which can be expected to have applications in general relativity[18]. Without loss of generality, if F can be written as the graph of the vector-valued function f , we may assume V is a unit timelike vector of the formV = {0, · · · , 0, vm+1, · · · , vm+n}. This can always be achieved by scaling F andthen choosing a suitable coordinate system [19]. Then the translating solitionssatisfy

⎧⎪⎨

⎪⎩

gij ∂2 f α

∂xi∂x j= vα, α = m + 1, · · · , m + n,

|Df (x)| < 1, ∀x ∈ Rm.

(7)


As in [11, 12], we shall consider the radially symmetric translating solitons

of the mean curvature flow (2). For this purpose, let r = |x| =√∑m

i=1(xi)2 bethe distance function in Rm. By direct computation, we find the induced metric(6) is given by

gij = δij − |Df |2r2

xix j,

where now |Df |2 = ∑m+nα=m+1 | f α|2 and f α = d

dr f α . The inverse of gij is

gij = δij + 1

1 − |Df |2|Df |2

r2xix j.

Therefore by (7), we have the following equations(

δij + 1

1 − |Df |2|Df |2

r2xix j

)∂2 f α

∂xi∂x j= vα, α = m + 1, · · · , m + n. (8)

The spacelike condition reads as

|Df (x)| =√√√√

m+n∑

α=m+1

| f α|2 < 1, ∀x ∈ Rm. (9)

Denote by v = (vm+1, · · · , vm+n), the constant vector in Rn, then |v| = 1.The following is the first main theorem in our paper

Theorem 1 There exists a vector-valued solution u(r) ∈ C2[0, ∞) to the initialvalue problem

⎧⎨

⎩

u1 − |u|2 + m − 1

ru = v,

u(0) = u(0) = 0,

(10)

such that f (x) = u(|x − x0|) + f (x0) in Rm for any radially symmetric C2 solu-tion f = ( f m+1, · · · , f m+n) of (8) and (9), where the vector-valued function u =(um+1, · · · , um+n), and u means ( dum+1

dr , · · · , dum+n

dr ). Moreover, the vector-valuedfunction u(r) ∈ C∞[0, ∞) satisf ies

r√m2 + r2

≤ |u(r)| ≤ e2r − 1

e2r + 1, ∀r ∈ [0, ∞) (11)

and

0 < |u(r)| � 1, ∀r ∈ [0, ∞). (12)

Next we consider the Euclidean space case. If F : Rm → Rm+n is an m-dimensional noncompact and complete immersed into a Euclidean spaceRm+n, then by a similar discussion as in the Minkowski space case, Fcan be written locally as a graph of f = ( f m+1, · · · , f m+n), and up to a

88 G. Li et al.

diffeomorphism in Rm, the radially symmetric translating soliton solutions withtranslating vector V satisfy the following equations

⎧⎪⎨

⎪⎩

(

δij − 1

1 + |Df |2|Df |2

r2xix j

)∂2 f α

∂xi∂x j= vα, α = m + 1, · · · , m + n,

|Df (x)| < ∞, ∀x ∈ Rm.

(13)

Theorem 2 There exists a vector-valued solution u(r) ∈ C2[0, ∞) to the initialvalue problem

⎧⎨

⎩

u1 + |u|2 + m − 1

ru = v,

u(0) = u(0) = 0,

(14)

such that f (x) = u(|x − x0|) + f (x0) in Rm for any radially symmetric C2 solu-tion f = ( f m+1, · · · , f m+n) of (13), where the vector-valued function u =(um+1, · · · , um+n), and u means ( dum+1

dr , · · · , dum+n

dr ). Moreover, the vector-valuedfunction u(r) ∈ C∞[0, ∞) satisf ies

0 ≤ |u(r)| <r

m − 1, ∀r ∈ [0, ∞) (15)

and

|u(r)| > 0, ∀r ∈ [0, ∞). (16)

When r → ∞, we further have the asymptotic expansion

|u(r)| = rm − 1

− 1

r+ o

(r−2

). (17)

Remark 1

(i) In Theorems 1 and 2, we do not know if the solution u(r) is unique. Butif there are two solutions u1 and u2 to (10), we see that v f1(x) = vu1(|x|)and v f2(x) = vu2(|x|) are solutions to the equation

⎧⎪⎨

⎪⎩

gij ∂2(v f )

∂xi∂x j= 1, |v| = 1,

|Df (x)| < 1, ∀x ∈ Rm.

This is a strictly elliptic equation on v f in any ball BR(0) ⊂ Rm. Thereforeby the uniqueness theorem in [5], we have v(u1 − u2) = 0.

(ii) By using the method in [6], we can obtain more accurate asymptoticexpansions of |u|.

For spacelike submanifolds, by integrating (11), we have√

m2 + r2 − m ≤ |u(r)| < r.

So the rotationally symmetric soliton solution tends to infinity linearly as r →∞. We can describe the asymptotic behavior of general solitons as |x| → ∞,


as that done of spacelike hypersurfaces by Jian [11], by using the tangentcones methods in [4, 21] for entire spacelike submanifolds of constant meancurvature. Define the blowdown of f according to the vector v at infinity by

V f (x) = limρ→∞

v f (ρx)

ρ, (18)

where v is a unit constant vector in Rn, and f (x) is a vector-valued function ofRm. If v f (x) is convex, then

ddρ

(v f (ρx)

ρ− v f (0)

ρ

)

≥ 0.

On the other hand,

v f (ρx)

ρ− v f (0)

ρ≤ |D(v f )||x| = |v · Df ||x| ≤ |Df ||x| ≤ |x|

since the graph of f is spacelike. Therefore if v f is a convex function satisfying(7), V f (x) is well-defined over Rm and the limit in (18) exists uniformly on anycompact set in Rm. We then have the following

Theorem 3 Suppose that f is a solution to (7) and v f is convex. Then theblowdown function V f is a positive homogeneous degree one convex functionsatisfying the following limits

V f (y) = limρ→∞

v f (ρy)

ρ= 1 uniformly for y ∈ ∇ f (Rm) ∩ Sm−1 (19)

and

V f (x) = limρ→∞

v f (ρx)

ρ= |x| for x ∈ ∇ f (Rm), (20)

where ∇ f (Rm) is the smallest closed set containing {y : y = ∇ f (x), x ∈ Rm}in Rm.

We remark that for solutions to (13), similar convexity can also be obtained.

3 Perturbation Equations

We start with some simple facts on radical symmetric solutions to equa-tions (8).

Given any x0 ∈ Rm, we assume f ∈ Ck,α(Rm) is a vector-valued function forsome k ≥ 1, 0 ≤ α ≤ 1 with k + α ≥ 2, and satisfy f (x) = u(|x − x0|) + f (x0).Then u(0) = 0. Moreover as that of spacelike hypersurfaces [11], we haveu(r) ∈ Ck,α[0, ∞) and u(0) = 0, here u(r) = du

dr . Thus the equations (8) are

90 G. Li et al.

equivalent to the following differential equation on vector-valued functionu = (um+1, · · · , um+n)

u(r)1 − |u(r)|2 + m − 1

ru(r) = v, |v| = 1, r ∈ (0, ∞) (21)

u(0) = u(0) = 0, (22)

and the spacelike condition is equivalent to

0 ≤ |u(r)| < 1, ∀r ∈ [0, ∞). (23)

Conversely, if u ∈ C2[0, ∞) is a solution to (21)–(23), then it follows fromdirect computation that f (x) = u(|x|) ∈ C1,1(Rm) is a solution to (8) and (9).In our setting, (7) is an elliptic system, we can not directly use the regularitytheory of elliptic equations, but we can use the regularity theory of a systemof elliptic equations in [13] to conclude that u(|x|) ∈ C∞(Rm) and thus u ∈C∞[0, ∞) (see Section 4 below).

Since (21) is singular at r = 0, we consider the approximation problem forany ε > 0

uε(r)1 − |uε(r)|2 + m − 1

r + εuε(r) = v, |v| = 1, r ∈ [0, ∞) (24)

|uε(r)| < 1, ∀r ∈ [0, ∞) (25)

uε(0) = 0, uε(0) = ε

mv. (26)

First (24) reads as

uεuε

1 − |uε|2 = vuε − m − 1

r + ε|uε|2.

Integrating the above equation from 0 to r > 0 we have

ln

√1 − |uε(0)|21 − |uε(r)|2 =

∫ r

0

(

vuε − m − 1

r + ε|uε|2

)

dr,

which implies for any r < ∞

ln

√1 − |uε(0)|21 − |uε(r)|2 < ∞

by (25) and (26). Thus for any R > 0, there exists a constant 0 < C(R) < 1depending only on R such that

|uε(r)| < 1 − C(R), ∀r ∈ [0, R).

Therefore by local existence theory of a system of ODEs, we see that for anyε ∈ (0, 1) there is a unique smooth solution to (24)–(26). Obviously,

uε(0) = (1 − |uε(0)|2)

(

v − m − 1

εuε(0)

)

= m2 − ε2

m3v,


and so

|uε(0)| = m2 − ε2

m3, and vuε(0) = m2 − ε2

m3> 0. (27)

The following is a key lemma in our paper

Lemma 1 The solutions to (24)–(26) have the following properties for ∀r ∈[0, ∞)

(i) |vuε(r)| = |uε(r)|;(ii) vuε(r) > 0;

(iii) uε(r)uε(r) ≥ 0.

Proof(i) First we remark that vuε(r) ≥ 0 for any r ∈ [0, ∞). This is because

vuε(0) > 0 and vuε(0) > 0 by (26) and (27). So vuε(r) ≥ 0 and vuε(r) ≥ 0 nearr = 0. If there exists an r0 > 0 such that vuε(r0) = 0 for the first time, then by(24), vuε(r0) > 0, which implies vuε(r) is strictly increasing at r0, and thereforewe prove vuε(r) ≥ 0 everywhere.

Next we consider the function

Z (r) = | cos ∠(v, uε)| = |vuε|2|uε|2 .

By using (24), direct computation shows that

ddr

Z = ddr

|vuε|2|uε|2 = 1

|uε|4{2vuε · vuε|uε|2 − 2|vuε|2uε · uε

}

= 2

|uε|4{

vuε|uε|2[(1 − |uε|2

)(

1 − m − 1

r + εvuε

)]

−|vuε|2[(1 − |uε|2

)(

vuε − m − 1

r + ε|uε|2

)]}

= 2

|uε|4 vuε

(1 − |uε|2

) (|uε|2 − |vuε|2) ≥ 0,

which implies Z is not decreasing of r. By (26), Z (0) = 1, and since 0 ≤ Z ≤ 1,we have Z (r) ≡ 1 and therefore

|vuε(r)| = |uε(r)|for any r ∈ [0, ∞). This proves (i).

(ii) We first prove the Claim: vuε(r) ≥ 0 for any r ∈ [0, ∞). Otherwise, thereis r1 ∈ (0, ∞) such that vuε(r1) < 0. Then we may choose some r0 > 0 and δ > 0such that

vuε(r0) = 0, vuε(r) < 0, ∀r ∈ (r0, r0 + δ),

92 G. Li et al.

and without loss of generality, we may assume vuε(r) ≥ 0 for any r ≤ r0. By(25) and (26), we may further choose δ small such that vuε(r) > 0, for ∀r ∈[r0, r0 + δ]. Hence

vuε(r0) > vuε(r) > 0, ∀r ∈ (r0, r0 + δ).

On the other hand by (24) again, for any r ∈ (r0, r0 + δ)

1 = m − 1

r0 + εvuε(r0) >

m − 1

r + εvuε(r)

>m − 1

r + εvuε(r) + vuε(r)

1 − |uε(r)|2 = 1

is a contradiction, hence the claim.By the claim, we have vuε(r) ≥ ε

m for any r ∈ [0, ∞), and we can now prove(ii).

If there exists r2 > 0 such that vuε(r2) = 0, we consider the function U(r) =m−1r+ε

vuε(r) = 1 − vuε(r)1−|uε(r)|2 , which attains its maximum 1 at r2. Then

U(r2) = 0 ⇐⇒ vuε(r2) = 0,

again a contradiction. This proves (ii).(iii) First we see uε(0)uε(0) = (m2−ε2)ε

m4 > 0. If there exists an r3 > 0 such thatuε(r3)uε(r3) < 0 for the first time, as before we may choose r0 > 0 and smallδ > 0 such that

uε(r0)uε(r0) = 0, and uε(r)uε(r) < 0, ∀r ∈ (r0, r0 + δ).

This implies |uε(r)|2 is strictly decreasing in (r0, r0 + δ). But by (ii), vuε(r) ispositive and not decreasing, and by (i), |uε(r)| is not decreasing, a contradiction.This completes the proof of the lemma. ��

Corollary 1 The solutions to (24)–(26) satisfy for any r ∈ [0, ∞)

(i) vuε(r) = |uε(r)| > 0;(ii) |uε(r)| > 0.

4 Proofs of Theorems

In this section, we first prove Theorem 1. We need the following lemmas

Lemma 2 There exists a solution u(r) ∈ C∞[0, ∞) to (10).

Proof We have proved by Lemma 1(i) and (ii) that

ε2

m2≤ |uε(r)|2 < 1. (28)


By Lemma 1 again, we have vuε(r) ≥ vuε(0) = εm , and therefore

|uε(r)| ≥ vuε(r) ≥ εrm

.

At the same time

ddr

|uε|2 = 2uεuε ≤ 2|uε|,which implies

|uε(r)| ≤ r, ∀r ∈ [0, ∞).

We then haveεrm

≤ |uε(r)| ≤ r, ∀r ∈ [0, ∞). (29)

By Lemma 1(i), we have, uε(r) = (vuε(r)) v. Then uε(r) has the form by (24)

uε(r) = (1−|uε(r)|2

)(

v− m − 1

r + εuε(r)

)

=(1−|uε(r)|2

)(

1− m − 1

r + ε(vuε(r))

)

v.

Thus by Lemma 1(ii), 1 − m−1r+ε

(vuε(r)) > 0, and so

0 < |uε(r)| = (1 − |uε(r)|2

)(

1 − m − 1

r + ε(vuε(r))

)

≤ 1 − m − 1

r + ε|uε(r)|

≤ 1 − (m − 1)ε

m(r + ε). (30)

By (28)–(30), the Arzela–Ascoli theorem implies that we can choose asubsequence εk → 0(k → ∞) and a function u(r) ∈ C1,α[0, R](α ∈ (0, 1) fixed)for any R > 0 such that

uεk → u in C1,α[0, R] as k → ∞. (31)

Obviously

u(0) = 0 = u(0), 0 ≤ |u(r)| ≤ 1,

and

0 ≤ |u(r)| ≤ 1, ∀r ∈ [0, ∞). (32)

Next, we prove 0 ≤ |u(r)| < 1 for any r ∈ [0, ∞). Otherwise, there is r4 > 0such that |u(r4)| = 1 and 0 ≤ |u(r)| < 1 for all r ∈ [0, t4). Integrating (24) foruεk uεk over [ r4

2 , r] we obtain

ln(1 − |uεk |2

) |rr42

= −2∫ r

r42

vuεk dr + 2(m − 1)

∫ r

r42

|uεk |2r + εk

dr,

94 G. Li et al.

or

ln(1 − |uεk(r)|2

) − ln(

1 − |uεk

(r4

2

)|2

)≥ −2

∫ r

r42

|uεk |dr ≥ −2(

r − r4

2

).

Let k → ∞ and r → t−4 , we have

−∞ − ln(

1 − |u(r4

2

)|2

)≥ −r4,

a contradiction.Since R > 0 is arbitrary, we have a solution u(r) ∈ C2[0, ∞) to (10). By

direct computation, f (x) = u(|x|) ∈ C1,1(Rm) is a solution to (7). Now bySchauder theory of elliptic system [13], we see that f (x) is smooth in anycompact set of Rm. Then by the discussion at the beginning of Section 3, u(r)is smooth, i.e. u ∈ C∞[0, ∞). This completes the proof of the lemma. ��

Lemma 3 If the vector-valued function u(r) ∈ C2[0, ∞) is a solution to (10),then it satisf ies (11) and (12).

Proof We first prove (12). By (32) we only need to prove |u(r)| > 0 for anyr ∈ [0, ∞). For this purpose, we have to prove

vu(r) > 0, for any r ∈ (0, ∞). (33)

In fact, we observe that vu(0) = 1m > 0 and vu(r) ≥ 0 for any r ∈ [0, ∞). Then

(33) follows from vu(0) = 0.If there is r5 ∈ (0, ∞) such that vu(r5) = 0, as the proof of Lemma 1(ii), we

get a contradiction by using (33). So |u(r)| ≥ vu(r) > 0 for any r ∈ [0, ∞).Next we prove (11). As vectors in Rn, the first equation of (9) can be

written as,

u = (1 − |Du|2)

(

v − m − 1

ru)

= (1 − |u|2)

(

v − m − 1

ru)

.

Integrate both sides of the above equality from r0 to r (0 < r0 < r) we have

u(r) − u(r0) = v

∫ r

r0

(1 − |u|2) dr −

∫ r

r0

(1 − |u|2) m − 1

rudr

which implies

|u(r)| + |u(r0)| ≥∫ r

r0

(1 − |u|2) dr −

∫ r

r0

(1 − |u|2) m − 1

r|u|dr. (34)

If |u(r)|2 ≤ 1 − δ for all r ∈ [0, ∞) and some δ ∈ (0, 1), then there existsr0 > 0 such that for all r > r0, δ < 1 − |u|2 < 1. It is easy to see that (1 −|u|2)m−1

r |u| < m−1r . Therefore by (34),

|u(r)| + |u(r0)| ≥ δ(r − r0) − (m − 1)(ln r − ln r0). (35)


It is easy to see that the right hand side of (35) tends to infinity as r → ∞,which is impossible. This shows that

limr→∞ |Du|2 = lim

r→∞ |u|2 = 1.

Since by Lemma 2, uε as well as its all derivatives converges to u uniformlyas ε → 0, we have by Corollary 1(i),

vu(r) = |u(r)|, and limr→∞ vu(r) = lim

r→∞ |u(r)| = 1. (36)

We then can prove the left inequality of (11). Let

W(r) = vu(r) − r√m2 + r2

.

If the first inequality of (11) is false, then by (36), there exists r6 > 0, suchthat W(r6) < 0. Obviously W(0) = 0 and by (36) again W(∞) = 0. We see thatW(r) attains its negative minimum at a point r7 > 0. Hence W(r7) = 0, i.e.,

vu(r7) = m2

(m2 + r2

7

) 32

.

On the other hand, by (10),

vu(r7) = (1 − |u(r7)|2

)(

1 − m − 1

r7vu(r7)

)

>

(

1 − r27

m2 + r27

)⎛

⎝1 − m − 1

r7

r7√

m2 + r27

⎞

⎠

= m2

m2 + r27

⎛

⎝1 − m − 1√

m2 + r27

⎞

⎠ .

The above two expressions give that m√m2+r2

7

> 1, a contradiction.

Now we prove the right inequality of (11). By (10) again,

vu ≤ 1 − (vu)2.

By direct calculation, we have 1+vu1−vu ≤ e2r, which implies vu ≤ e2r−1

e2r+1 . This com-pletes the proof of the Lemma. ��

Combining Lemma 2 and 3, we finish the proof of Theorem 1.

Proof of Theorem 2 As in the spacelike case, a vector-valued solution f ∈Ck,α(Rm) to (13) satisfying f (x) = u(|x − x0|) + f (x0) is equivalent to a

96 G. Li et al.

solution u(r) ∈ Ck,α[0, ∞) to (14). Clearly, this equation is singular at r = 0,as before we consider the perturbed equation for any ε > 0

uε(r)1 + |uε(r)|2 + m − 1

r + εuε(r) = v, |v| = 1, r ∈ [0, ∞)

|uε(r)| < ∞, ∀r ∈ [0, ∞)

uε(0) = 0, uε(0) = ε

mv.

Then Lemma 1 still holds for this case, and we have a sequence εk → 0(k →∞) and a function u(r) ∈ C1,α[0, ∞)(α ∈ (0, 1) fixed) such that

uεk → u in C1,α[0, ∞) as k → ∞.

Moreover the vector-valued function u(r) is a solution to (14), and satisfies (15)and (16).

Let ϕ(r) = vu(r), then it satisfies the equation

ϕ(r) = (1 + ϕ2

)(

1 − m − 1

rϕ(r)

)

.

By this equation, the asymptotic expansion (17) can be established as that in[3]. We thus finish the proof of Theorem 2.2. ��

Proof of Theorem 3 Our proof is essentially same as that in [11]. For complete-ness, we outline the proof here. We now assume f = ( f m+1(x), · · · , f m+n(x))

(x ∈ Rm) is a vector-valued function on Rm, then (7) can be rewritten as⎧⎪⎨

⎪⎩

gij ∂2 f (x)

∂xi∂x j= v, |v| = 1,

|Df (x)| < 1, ∀x ∈ Rm.

(37)

From the definition of V f (x), we see that it is convex if v f is so. And for anyλ > 0

V f (λx) = limρ→∞

v f (ρλx)

ρ= lim

ρ→∞v f (ρλx)

ρλλ = λV f (x),

so V f (x) is a positively homogeneous degree one function.

For any x, y ∈ Rm, we have by (37) that

|V f (x) − V f (y)| ≤ lim supρ→∞

|v f (ρx) − v f (ρy)|ρ

≤ |D(v f )||x − y| ≤ |x − y|.

Lemma 4 For any x ∈ Rm, and any δ > 0, there exists a y ∈ Rm such that

|V f (x) − V f (y)| = |x − y| = δ. (38)

Proof If (38) is false, there would exist an x ∈ Rm, δ > 0 and θ > 0 such that

V f (y) ≤ V f (x) + (1 − 2θ)δ


for all y ∈ Rm, with |x − y| = δ. Observing that the limit (18) is uniform on anycompact set, we may choose a ρ0 > 0 so that

v fρ(y) ≤ V f (x) + (1 − θ)δ, (39)

for all ρ > ρ0, and all y ∈ B(x, δ), where B(x, δ) = {y ∈ Rm : |y − x| < δ}, andv fρ(y) = v f (ρy)

ρ.

It follows from (37) that v fρ satisfies⎧⎪⎨

⎪⎩

gij ∂2

∂xi∂x j(v fρ(x)) = ρ, |v| = 1,

|D(v fρ(x))| ≤ |Df (x)| < 1, ∀x ∈ Rm,

(40)

where gij is the inverse of gij = δij − ∂ f α

∂xi∂ f α

∂xi .If u(|x|) is a radical symmetric solution to (37) as in Theorem 1, then the

function

w(y) = w(y; ρ) := V f (x) +(

δ − vu(ρδ)

ρ+ vu(ρ|y − x|)

ρ

)

− θδ

is also a solution to (40) as v fρ for any ρ > 0, and x ∈ Rm. Obviously wheny ∈ ∂ B(x, δ),

w(y) = V f (x) + (1 − θ)δ.

So by (39), applying the maximum principle for elliptic equations to v fρ(y) −w(y; ρ), we have on B(x, δ)

v fρ(y) ≤ w(y; ρ), ∀y ∈ B(x, δ).

Then let ρ → ∞, we have

V f (y) = limρ→∞

v f (ρy)

ρ≤ V f (x) +

(

δ − limρ→∞

vu(ρδ)

ρ+ lim

ρ→∞vu(ρ|y − x|)

ρ

)

− θδ

= V f (x) + (1 − θ)δ + (|y − x| − δ) = V f (x) + |y − x| − θδ.

Here we use the fact that√

m2 + r2 − m ≤ vu(r) ≤ r

by the proof of Theorem 1. Take y = x, we have

V f (x) ≤ V f (x) − θδ,

a contradiction. This proves the lemma. ��

We have seen that V f (x) : Rm → R is a convex function if v f is convex.Recall that the tangential mapping of V f at a point x0 ∈ Rm is defined by

TV f (x0) = {α ∈ Rm : V f (x) ≥ α(x − x0) + V f (x0), ∀x ∈ Rm}

.

98 G. Li et al.

It is a closed, convex set and equals to DV f (x0) if V f is differential at x0.The tangent cone of V f is defined by

TV f

(Rm) = ∪x∈Rm TV f (x).

The following is also a lemma in [11].

Lemma 5 If f is a vector-valued function satisfying (37), and v f is convex, thenthe tangent cone of V f satisf ies

TV f (Rm) = TV f (0) = D(v f )(Rm).

Using Lemmas 4 and 5, similarly as in [11], we can get the limits (19) and(20). Therefore we finish the proof of Theorem 3.

References

1. Alias, L., Romero, A., Sanchez, M.: Uniqueness of complete spacelike hypersurfaces of con-stant mean curvature in generalized Robertson–Walker spacetimes. Gen. Relat. Gravit. 27,71–84 (1995)

2. Chen, J., Li, J.: Mean curvature flow of surface in 4-manifolds. Adv. Math. 163(2), 287–309(2001)

3. Clutterbuck, J., Schnürer, O., Schulze, F.: Stability of translating solutions to mean curvatureflow. Calc. Var. Partial Differ. Equ. 29, 281–293 (2007)

4. Choi, H., Treibergs, A.: Gauss maps of spacelike constant mean curvature hypersurfaces ofMinkowski space. J. Differ. Geom. 32 775–817 (1990)

5. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd edn.Springer-Verlag (1983)

6. Gui, C., Jian, H., Ju, H.: Properties of translating solutions to mean curvature flow. DiscreteContin. Dyn. Syst. 28(2), 441–453 (2010)

7. Han, X., Li, J.: Translating solitons to symplectic and Lagrangian mean curvature flows. Int. J.Math. 20(4), 443–458 (2009)

8. Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1),237–266 (1984)

9. Huisken, G., Sinestrari, C.: Mean curvature flow singularities for mean convex surfaces. Calc.Var. Partial Differ. Equ. 8(1), 1–14 (1999)

10. Hungerbhler, N., Smoczyk, K.: Soliton solutions for the mean curvature flow. Differ. IntegralEqu. 13(10–12), 1321–1345 (2000)

11. Jian, H.: Translating solitons of mean curvature flow of noncompact spacelike hypersurfacesin Minkowski space. J. Differ. Equ. 220(1), 147–162 (2006)

12. Jian, H, Liu, Q., Chen, X.: Convexity and symmetry of translating solitons in mean curvatureflows. Chin. Ann. Math. Ser. B, 26(3), 413–422 (2005)

13. Jost, J.: Nonlinear Methods in Riemannian and Kahlerian Geometry, DMV Seminar, vol. 10.Birkhuser Verlag, Basel (1988)

14. Joyce, D., Lee, Y., Tsui, M.: Self-similar solutions and translating solitons for Lagrangian meancurvature flow. J. Differential Geom. 84(1), 127–161 (2010)

15. Li, G., Salavessa, I.: Mean curvature flow and Bernstein–Calabi results for spacelike graphs.Differential Geometry, pp. 164–174. World Sci. Publ., Hackensack, NJ (2009)

16. Li, G., Salavessa, I.: Graph Bernstein results in cruved pseudo-Riemannian manifolds. J.Geom. Phys. 59, 1306–1313 (2009)

17. Li G., Salavessa I.: Mean curvature flow of spacelike graphs. Math. Z. (2010). doi:10.1007/s00209-010-0768-4

18. Marsden, J., Tipler, F.: Maximal hypersurfaces and foliations of constant mean curvature ingeneral relativity. Phys. Rep. 66(3), 109–139 (1980)

http://dx.doi.org/10.1007/s00209-010-0768-4

http://dx.doi.org/10.1007/s00209-010-0768-4


19. Neves, A., Tian, G.: Translating solutions to Lagrangian mean curvature flow. (math.DG)arXiv:0711.4341 (2007)

20. Smoczyk, K.: Angle theorems for the Lagrangian mean curvature flow. Math. Z. 240(4), 849–883 (2002)

21. Treibergs, A.: Entire spacelike hypersurfaces of constant mean curvature in Minkowski space.Invent. Math. 66, 39–56 (1982)

22. Wang, M.: Mean curvature flow of surfaces in Einstein four-manifolds. J. Differ. Geom. 57,301–338 (2001)

23. Wang, M.: Long-time existence and convergence of graphic mean curvature flow in arbitrarycodimension. Invent. Math. 148(3), 525C–543 (2002)

24. Wang, M.: The Dirichlet problem for the minimal surface system in arbitrary dimensions andcodimensions. Commun. Pure Appl. Math. 57(2), 267–281 (2004)

25. Xin, Y.: Mean curvature flow with convex Gauss image. Chin. Ann. Math. Ser. B 29(2), 121–134 (2008)

http://arXiv.org/abs/0711.4341

Math Phys Anal Geom (2011) 14:101–114DOI 10.1007/s11040-011-9089-z

Persistence Properties and Unique Continuationof Solutions to a Two-component Camassa–HolmEquation

Zhengguang Guo · Lidiao Ni

Received: 17 January 2010 / Accepted: 13 January 2011 / Published online: 22 January 2011© Springer Science+Business Media B.V. 2011

Abstract We will consider a two-component Camassa–Holm system whicharises in shallow water theory. The present work is mainly concerned withpersistence properties and unique continuation to this new kind of system, inview of the classical Camassa–Holm equation. Firstly, it is shown that thereare three results about these properties of the strong solutions. Then we alsoinvestigate the infinite propagation speed in the sense that the correspondingsolution does not have compact spatial support for t > 0 though the initial databelongs to C∞

0 (R).

Keywords Two-component Camassa–Holm equation ·Persistence properties · Propagation speed

Mathematics Subject Classifications (2010) 37L05 · 35Q58 · 26A12

1 Introduction

This work is concerned with the following integrable model named two-component Camassa–Holm system:{

ut − uxxt + 3uux − 2uxuxx − uuxxx + σρρx = 0, t > 0, x ∈ R,ρt + (ρu)x = 0, t > 0, x ∈ R,

(1.1)

Z. GuoDepartment of Mathematics, East China Normal University, Shanghai 200241, Chinae-mail: [email protected]

L. Ni (B)Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, Chinae-mail: [email protected]

102 Z. Guo, L. Ni

where u(x, t) describes the horizontal velocity of the fluid and ρ(x, t) describesthe horizontal deviation of the surface from equilibrium, all measured indimensionless units. σ can be chosen 1 or −1. We require that u(x, t) and ρ(x, t)decay rapidly at infinity.

This system appears originally in [30] and its mathematical properties havebeen studied further in many works, e.g. [12, 18–20, 23]. Recently, Guo andZhou in [23] analyzed a wave breaking mechanism and the global existence ofsolutions. While Guo discussed global existence and blow up phenomena forthis kind of equation via the associated potential in [22]. The case σ = −1 cor-responds to the situation in which the gravity acceleration points upwards [12].The system with σ = −1 has been recently proposed by Chen et al. in [18]and Falqui in [20]. Henry proved the infinite propagation speed for this systemrecently in [25].

Set Q = (1 − ∂2x), then the operator Q−1 in R can be expressed by

Q−1 f = G ∗ f = 1

2

∫R

e−|x−y| f (y)dy,

where the sign ∗ denotes the spatial convolution, G is the associated Green’sfunction of the operator Q−1. So system (1.1) can be written as⎧⎨

⎩ut + uux + ∂xG ∗

(u2 + 1

2u2

x + σ

2ρ2

)= 0, t > 0, x ∈ R;

ρt + uρx = −uxρ, t > 0, x ∈ R.(1.2)

In addition, the model is also an integrable system in the sense that it hasLax-pair [12], the associated spectral parameter yields the following twoconservation laws:∫

R

(u2 + u2

x + σρ2)

dx and∫

R

(u3 + uu2

x + σuρ2)

dx,

which have been precisely proved in [23] by using the classical energy method.Very recently, smooth traveling wave solutions with σ = 1 was investigated byMustafa in [29]. In [12], Constantin and Ivanov gave a demonstration aboutits derivation in view of shallow water theory from the hydrodynamic point ofview.

Obviously, if ρ ≡ 0, system (1.1) reduces to the Camassa–Holm equation

ut − uxxt + 3uux = 2uxuxx + uuxxx, t > 0, x ∈ R, (1.3)

which was derived physically by Camassa and Holm in [17] by approximatingdirectly the Hamiltonian for Euler’s equations in the shallow water regime,where u(x, t) represents the free surface above a flat bottom. It is an integrableinfinite-dimensional Hamiltonian system, see [4, 11, 14]. It also appears inthe context of hereditary symmetries studied by Fuchssteiner and Fokas [21].Some satisfactory results have been obtained recently [6, 8, 12, 13, 17, 27, 28]for strong solutions. Moreover, wave breaking for a large class of initial datahas been established in [5, 8, 9, 32, 34, 35]. In [33], Xin and Zhang showedglobal existence and uniqueness for weak solutions with u0(x) ∈ Hs such that

Persistence Properties and Unique Continuation of Solutions... 103

u0(x) − u0xx(x) is a sign-definite Random measure, which also seen in [2, 3, 15].The solitary waves of Camassa–Holm equation are peaked solitons and areorbitally stable [16] (see also [1, 7, 10, 31, 32, 36]). It is worthy of beingmentioned here is the unique continuation and infinite propagation speed ofsolutions to the Camassa–Holm equation, which was presented by Zhou andhis collaborators in their work [26].

Here we mainly discuss the case of ρ(x, t) �= 0. The main purpose of thispaper is to investigate the persistence properties and unique continuation ofsolutions to (1.1) with the case of σ = 1. Some of our results are motivated bythe recent works in [23, 26, 37] and other relevant literature, such as [12, 25]and so on.

Due to the similarity of (1.3), just by following the argument for theCamassa–Holm equation, it is easy to establish the following well-posednesstheorem for (1.3)

Theorem 1.1 Given X0 = (u0, ρ0)T ∈ Hs × Hs−1, s > 2, then there exists a max-

imal T = T(‖X0‖Hs×Hs−1) > 0, and a unique solution X = (u, ρ)T to system(1.1) such that

X = X(·, X0) ∈ C([0, T); Hs × Hs−1) ∩ C1([0, T); Hs−1 × Hs−2).

Moreover, the solution depends continuously on the initial data, i.e. the mapping

X → X(·, X0) : Hs × Hs−1 →C([0, T); Hs × Hs−1) ∩ C1([0, T); Hs−1 × Hs−2)

is continuous.

In [19], the authors gave a detailed description on this well-posednesstheorem by Kato’s semi-group theory.

We now finish this introduction by outlining the rest of this paper. InSection 2, we show the persistence properties and unique continuation ofsolutions to the system (1.1). Then we turn our attention to the infinitepropagation speed on the whole line case in Section 3.

2 Persistence properties and unique continuation

In this section, we shall investigate the following properties for the strongsolutions to (1.1) (or system (1.2)) in L∞−space. The main idea comes from arecent work of Zhou and his collaborators [26].

Theorem 2.1 Assume that for some T > 0, s > 52 , and, σ = 1, X = (u, ρ)T ∈

C([0, T); Hs × Hs−1) is a strong solution of the initial value problem associatedto system (1.1), and that u0(x) = u(x, 0), ρ0(x) = ρ(x, 0) satisfy

|u0(x)|, |∂xu0(x)|, |ρ0(x)| ∼ O(e−θx) as x → ∞

104 Z. Guo, L. Ni

for some θ ∈ (0, 1). Then

|u(x, t)|, |ux(x, t)| ∼ O(e−θx) as x → ∞uniformly in the time interval [0, T].

Proof The proof is organized as follows. Firstly, we will give out the estimateson ‖u(x, t)‖L∞ . Then using the same method, we also can estimate the term‖ux(x, t)‖L∞ . Finally, we apply the weight function to obtain the desired result.Before giving out the proof, we introduce the following notation:

Notation

|u(x, t)| ∼ O(e−θx) as x → ∞, if limx→∞

|u(x)|e−θx

= L,

and

|u(x, t)| ∼ o(e−θx) as x → ∞, if limx→∞

|u(x)|e−θx

= 0.

Step 1 Estimate for ‖u(x, t)‖L∞ .

Multiplying the first equation in (1.2) by u2n−1 with n ∈ Z+, then integrating

both sides with respect to x− variable, we can get∫

R

u2n−1utdx +∫

R

u2n−1uuxdx +∫

R

u2n−1∂xG(x) ∗(

u2 + 1

2u2

x + 1

2ρ2

)dx = 0.

(2.1)The first term of the above identity is

∫R

u2n−1utdx = 1

2nddt

‖u(t)‖2nL2n = ‖u(t)‖2n−1

L2n

ddt

‖u(t)‖L2n ,

and the estimates of the second term is∣∣∣∣∫

R

u2n−1uuxdx

∣∣∣∣ � ‖ux(t)‖L∞‖u(t)‖2nL2n .

In view of Hölder’s inequality, we can obtain the following estimate for the lastterm in (2.1):

∫R

u2n−1∂xG(x) ∗(

u2 + 1

2u2

x + 1

2ρ2

)dx

� ‖u(t)‖2n−1L2n

∥∥∥∥∂xG(x) ∗(

u2 + 1

2u2

x + 1

2ρ2

)∥∥∥∥L2n

,

putting all the inequality above into (2.1) yield

ddt

‖u(t)‖L2n � ‖ux(t)‖L∞‖u(t)‖L2n +∥∥∥∥∂xG(x) ∗

(u2 + 1

2u2

x + 1

2ρ2

)∥∥∥∥L2n

.


Thus, using the Sobolev embedding theorem, there exists a constant

M = supt∈[0,T]

||u(x, t)||Hs

such that applying Gronwall’s gives us

‖u(t)‖L2n �(

‖u(0)‖L2n +∫ t

0

∥∥∥∥∂xG(x) ∗(

u2 + 1

2u2

x + 1

2ρ2

)∥∥∥∥L2n

dτ

)eMt. (2.2)

For any f ∈ L1(R) ∩ L∞(R), we know that

limq→∞ ‖ f‖Lq = ‖ f‖L∞ . (2.3)

Taking the limits in (2.2) (notice that G ∈ L1 and u2 + 12 u2

x + 12ρ2 ∈ L1 ∩ L∞)

from (2.3) we get

‖u(t)‖L∞ �(

‖u(0)‖L∞ +∫ t

0

∥∥∥∥∂xG(x) ∗(

u2 + 1

2u2

x + 1

2ρ2

)∥∥∥∥L∞

dτ

)eMt. (2.4)

Step 2 Estimate for ‖ux(x, t)‖L∞ .

In this step, we will establish an estimate on ‖ux(x, t)‖L∞ using the samemethod as above. Differentiating the first equation in (1.2) with respect to xvariable produces the following equation:

uxt + uuxx + u2x + ∂2

x G ∗(

u2 + 1

2u2

x + 1

2ρ2

)= 0. (2.5)

Again, multiplying (2.5) by u2n−1x with n ∈ Z

+, integrating the result in thex−variable, and considering the second term in the above identity with inte-gration by parts, one gets

∫R

uuxxu2n−1x dx =

∫R

u(

u2n−1x

2n

)x

dx = − 1

2n

∫R

uxu2nx dx,

so we have ∫R

uxtu2n−1x dx − 1

2n

∫R

uxu2nx dx +

∫R

u2n+1x dx

+∫

R

u2n−1x ∂2

x G ∗(

u2 + 1

2u2

x + 1

2ρ2

)dx = 0.

Similarly, the following inequality is true:

ddt

‖ux(t)‖L2n � 2‖ux(t)‖L∞‖ux(t)‖L2n +∥∥∥∥∂2

x G ∗(

u2 + 1

2u2

x + 1

2ρ2

)(t)

∥∥∥∥L2n

,

and therefore as before, we obtain

‖ux(t)‖L2n �(

‖u(0)‖L2n +∫ t

0

∥∥∥∥∂2x G(x) ∗

(u2 + 1

2u2

x + 1

2ρ2

)(τ )

∥∥∥∥L2n

dτ

)eMt.

(2.6)

106 Z. Guo, L. Ni

Taking the limits in (2.6) to obtain

‖ux(t)‖L∞ �(

‖u(0)‖L∞ +∫ t

0

∥∥∥∥∂2x G(x) ∗

(u2 + 1

2u2

x + 1

2ρ2

)(τ )

∥∥∥∥L∞

dτ

)eMt.

(2.7)

Step 3 Use the weight function to get the desired result.

We shall now introduce the weight function ϕN(x) with N ∈ Z+, which is

independent on t as follows:

ϕN(x) =⎧⎨⎩

1, x � 0,eθx, x ∈ (0, N),eθ N, x � N,

From the first equation in (1.2) and (2.5), we get the following twoequations:

ϕNut + ϕNuux + ϕN∂xG ∗(

u2 + 1

2u2

x + σ

2ρ2

)= 0,

ϕNuxt + ϕNuuxx + ϕNu2x + ϕN∂2

x G ∗(

u2 + 1

2u2

x + 1

2ρ2

)= 0.

We need some tricks to deal with the following term as in [37]:∫R

(ϕN)2n−1u2n−1ϕNuxdx =∫

R

(ϕNu)2n−1[(uϕN)x − u(ϕN)x]dx

=∫

R

(ϕNu)2n−1d(ϕNu) −∫

R

(ϕNu)2n−1u(ϕN)xdx

�∫

R

(ϕNu)2ndx,

where we use the fact 0 � ϕ′N(x) � ϕN(x), x ∈ R. Similar technique is used for

the term∫

R(ϕN)2n−1u2n−1

x ϕNuxxdx. Hence, as in the weightless case, we get thefollowing inequality in view of (2.4) and (2.7):

‖u(t)ϕN‖L∞ + ‖∂xu(t)ϕN‖L∞

� e2Mt (‖u(0)ϕN‖L∞ + ‖ux(0)ϕN‖L∞)

+ e2Mt(∫ t

0

∥∥∥∥ϕN∂xG(x) ∗(

u2 + 1

2u2

x + 1

2ρ2

)(τ )

∥∥∥∥L∞

+∥∥∥∥ϕN∂2

x G(x) ∗(

u2 + 1

2u2

x + 1

2ρ2

)(τ )

∥∥∥∥L∞

dτ

).

On the other hand, a simple calculation shows that there exists C > 0,depending only on θ ∈ (0, 1) such that for any N ∈ Z

+,

ϕN(x)

∫R

e−|x−y| 1

ϕN(y)dy � C.


Therefore for any appropriate function g, one sees that

∣∣ϕN∂xG(x) ∗ g2(x)∣∣ =

∣∣∣∣1

2ϕN(x)

∫R

e−|x−y|g2(y)dy

∣∣∣∣� 1

2ϕN(x)

∫R

e−|x−y| 1

ϕN(y)ϕN(y)g(y)g(y)dy

� 1

2

(ϕN(x)

∫R

e−|x−y| 1

ϕN(y)dy

)‖gϕN‖L∞‖g‖L∞

� C‖gϕN‖L∞‖g‖L∞ ,

and similarly,∣∣ϕN∂2

x G(x) ∗ g2(x)∣∣ � C‖gϕN‖L∞‖g‖L∞ .

In addition, denote

M = supt∈[0,T]

||ρ(x, t)||Hs−1 .

Therefore, it follows that there exists a constant C0, which depends onM, H, T, such that

‖u(t)ϕN‖L∞ + ‖ux(t)ϕN‖L∞

� C0 (‖u(0)ϕN‖L∞ + ‖ux(0)ϕN‖L∞)

+ C0

∫ t

0(‖u(τ )‖L∞ + ‖ux(τ )‖L∞ + ‖ρ(τ)‖L∞)

× (‖ϕNu(τ )‖L∞ + ‖ϕNux(τ )‖L∞ + ‖ϕNρ(τ)‖L∞) dτ

� C0(‖u(0)ϕN‖L∞ + ‖ux(0)ϕN‖L∞ + ‖ρ(0)ϕN‖L∞

+∫ t

0(‖ϕNu(τ )‖L∞ + ‖ϕNux(τ )‖L∞ + ‖ϕNρ(τ)‖L∞) dτ).

Hence, the following inequality is obtained for any N ∈ Z+ and any t ∈ [0, T]

‖u(t)ϕN‖L∞ + ‖ux(t)ϕN‖L∞

� C0 (‖u(0)ϕN‖L∞ + ‖ux(0)ϕN‖L∞ + ‖ρ(0)ϕN‖L∞)

� C0(‖u(0) max(1, eθx)‖L∞ +‖ux(0) max(1, eθx)‖L∞ +‖ρ(0) max(1, eθx)‖L∞

).

Finally, taking the limit as N goes to infinity in the above inequality, we canfind that for any t ∈ [0, T]

‖u(x, t)eθx‖L∞ + ‖ux(x, t)eθx‖L∞

� C0(‖u(0) max(1, eθx)‖L∞ +‖ux(0) max(1, eθx)‖L∞ +‖ρ(0) max(1, eθx)‖L∞

),

which completes the proof of Theorem 2.1. �

108 Z. Guo, L. Ni

The following theorem is to formulate decay conditions on a solution, attwo distinct times, which guarantee that X ≡ (0, ρ)T is the unique solution ofsystem (1.1).



|u0(x)| ∼ o(e−x), |u0x(x)|, |ρ0(x)| ∼ O(e−αx) as x → ∞,

for some α ∈ ( 12 , 1) and there exists t1 ∈ (0, T] such that

|u(x, t1)| ∼ o(e−x) as x → ∞,

then u ≡ 0, i.e. X ≡ (0, ρ)T.

Remark 2.1 Theorem 2.2 holds with the corresponding decay hypothesisstated for x < 0. The time interval [0, T] is the maximal existence time intervalof the strong solution. This guarantees that the solution is uniformly boundedin the Hs-norm in this interval, and that our solution is the strong limit ofsmooth ones such that the integration by parts in the proof can be justified.

Proof We shall prove this theorem following the result in Theorem 2.1. Firstly,integrating the first equation in (1.2) over the time interval [0, t1] we get

u(x, t1) − u(x, 0)

=∫ t1

0uux(x, τ )dτ +

∫ t1

0∂xG(x) ∗

(u2+ 1

2u2

x + 1

2ρ2

)(x, τ )dτ = 0.

(2.8)

According to the hypothesis and Theorem 2.1, it follows that

u(x, t1) − u(x, 0) ∼ o(e−x) as x → ∞, (2.9)

and ∫ t1

0uux(x, τ )dτ ∼ O(e−x) as x → ∞. (2.10)

If u(x, t) �= 0, the following deduction tells us the last term of (2.8) isinfinitesimal with the same order not higher order of e−x. Thus, a contradictionoccurs. ∫ t1

0∂xG(x) ∗

(u2 + 1

2u2

x + 1

2ρ2

)(x, τ )dτ

= ∂xG(x) ∗∫ t1

0

(u2 + 1

2u2

x + 1

2ρ2

)(x, τ )dτ = ∂xG(x) ∗ m(x).

Using the conditions and Theorem 2.1, we know

0 � m(x) ∼ O(e−2αx), so that m(x) ∼ o(e−x) as x → ∞. (2.11)


Therefore,

∂xG(x) ∗ m(x) = −1

2

∫R

sgn(x − y)e−|x−y|m(y)dy

= −1

2e−x

∫ x

−∞eym(y)dy + 1

2ex

∫ ∞

xe−ym(y)dy.

From (2.11) it follows that

e−x∫ x

−∞eym(y)dy=e−x

∫ x

−∞eyo(ey)dy=o(1)e−x

∫ x

−∞e2ydy∼o(1)e−x ∼o(e−x),

as x → ∞. If m �= 0, there exists a constant C1, such that∫ x

−∞eym(y)dy � C1, for x large enough.

Hence, we have

−∂xG(x) ∗ m(x) � C1

2e−x, for x large enough,

which combined with (2.8)–(2.10) yields a contradiction. Thus, m(x) ≡ 0 andconsequently u ≡ 0. The theorem is proved. �



|u0(x)| ∼ O(e−x), |u0x(x)|, |ρ0(x)| ∼ O(e−αx) as x → ∞,

for some α ∈ ( 12 , 1). Then

|u(x, t1)| ∼ O(e−x) as x → ∞,

uniformly in the time interval [0, T].

This proof is similar to that given for Theorem 2.2 and therefore it will beomitted.

3 Propagation speed

Recently, Zhou with his collaborators [26] proved the infinite propagationspeed for the Camassa–Holm equation by establishing a detailed descriptionon the profile of the corresponding solution with compactly supported initialdatum. It generalized earlier results in this direction obtained in [6, 24].

But how about the two-component generalized Camassa–Holm equation?Do this kind of equation have the same property as the Camassa–Holmequation? Fortunately, the answer is positive.

110 Z. Guo, L. Ni

It is worth to notice the equivalent form of the first equation in (1.1) asfollowing

yt + 2yux + yxu + ρρx = 0. (3.1)

Moreover, we also need to introduce the standard particle trajectory methodfor later use. Suppose u(x, t) solves the system (1.1), q(x, t) satisfies thefollowing initial value problem:{

qt = u(q, t), 0 < t < T, x ∈ R;q(x, 0) = x, x ∈ R,

(3.2)

where T is the lifespan of the solution, then q is a diffeomorphism of the line.Moreover, we know that the map q(·, t) is an increasing diffeomorphism of R

with

qx(x, t) = exp

(∫ t

0ux(q, s)ds

)> 0, (x, t) ∈ R × [0, T).

The important work in this section is to give a more detailed descriptionon the corresponding strong solution X(x, t) in its lifespan with X0 beingcompactly supported. The main theorem reads:

Theorem 3.1 Assume that the initial datum 0 �≡ X0(x) = (u0, ρ0)T ∈ Hs × Hs−1

with s > 52 is compactly supported in [a, c] with u0 �≡ 0, then the correspond-

ing solution X(x, t) = (u, ρ)T to system (1.1) has the following property: for0 < t < T,

u(x, t) = L(t)e−x as x > q(c, t); u(x, t) = l(t)ex as x < q(a, t),

with L(t) > 0 and l(t) < 0 for t ∈ (0, T] respectively, where q(x, t) is def inedby (3.2) and T is its lifespan. Furthermore, L(t) and l(t) denote continuousnonvanishing functions, with L(t) being a strictly increasing function, while l(t)being strictly decreasing.

Remark 3.1 This theorem implies that the strong solution does NOT havecompact x-support for any t > 0 in its lifespan, although the correspondingX0(x) is compactly supported.

Remark 3.2 Although the infinite propagation speed for system (1.1) has beenproved by Henry in his recent work [25], we discuss this property here simply,just for a comparison.

Proof First, since X0(x) has compact support, so do u0(x), y0(x) = (1 −∂2

x)u0(x) and ρ0(x). Applying particle trajectory method to (3.2) and the secondequation in system (1.1), we obtain

ddt

(y(q(x, t), t)q2

x(x, t)) = (yt + 2yux + yxu) (q(x, t), t))q2

x(x, t)

= −ρ(q(x, t), t))ρx(q(x, t), t))q2x(x, t)


and

ddt

(ρ(q(x, t), t)qx(x, t)) = 0,

which implies that

(ρ(q(x, t), t)qx(x, t)) = ρ0(x),

so we know that ρ is compactly supported in [q(a, t), q(c, t)] in its lifespan, i.e.ρ(x, t) = 0 for x > q(c, t) or x < q(a, t). Then we know when x > c or x < a

ddt

(y(q(x, t), t)q2

x(x, t)) = 0.

Thus y(q(x, t), t)q2x(x, t) is independent on time t over the interval (−∞, a) ∩

(c, ∞). We will get by taking t = 0 without loss of generality,

y(q(x, t), t)q2x(x, t) = y0(x) = 0, as x ∈ (−∞, a) ∩ (c, ∞).

This tells us that y(q(x, t), t) = 0 when x ∈ (−∞, a) ∩ (c, ∞), i.e. y is compactlysupported in [q(a, t), q(c, t)] in its lifespan. Hence the following functions arewell-defined:

E(t) =∫

R

ex y(x, t)dx and F(t) =∫

R

e−x y(x, t)dx, (3.3)

with

E(0) =∫

R

ex y0(x)dx =∫

R

exu0(x)dx −∫

R

exu0xx(x)dx = 0.

and F(0) = 0 by integration by parts.Then for x > q(c, t), we have

u(x, t) = 1

2e−|x| ∗ y(x, t) = 1

2e−x

∫ q(c,t)

q(a,t)eξ y(ξ, t)dξ = 1

2e−x E(t), (3.4)

where (3.3) is used.Similarly, when x < q(a, t), we get

u(x, t) = 1

2e−|x| ∗ y(x, t) = 1

2ex

∫ q(c,t)

q(a,t)e−ξ y(ξ, t)dξ = 1

2ex F(t). (3.5)

Because y(x, t) has compact support in x in the interval [q(a, t), q(c, t)] for anyt ∈ [0, T], we get y(x, t) = u(x, t) − uxx(x, t) = 0, for x > q(c, t) or x < q(a, t).Hence, as consequences of (3.4) and (3.5), we have

u(x, t) = −ux(x, t) = uxx(x, t) = 1

2e−x E(t), as x > q(c, t) (3.6)

and

u(x, t) = ux(x, t) = uxx(x, t) = 1

2ex F(t), as x < q(a, t). (3.7)

112 Z. Guo, L. Ni

On the other hand,

dE(t)dt

=∫

R

ex yt(x, t)dx.

Differentiating the first equation in (1.2) twice, we get

0 = uxxt + (uux)xx + ∂x∂2x G ∗

(u2 + 1

2u2

x + 1

2ρ2

)

= uxxt + (uux)xx − ∂x(1 − ∂2

x

)G ∗

(u2 + 1

2u2

x + 1

2ρ2

)

+∂xG ∗(

u2 + 1

2u2

x + 1

2ρ2

)

= uxxt + (uux)xx − ∂x

(u2 + 1

2u2

x + 1

2ρ2

)+ ∂xG ∗

(u2 + 1

2u2

x + 1

2ρ2

).

(3.8)

Combining the first equation in (1.2) and (3.8), we obtain

yt = (1 − ∂2

x

)ut = −uux + (uux)xx − ∂x

(u2 + 1

2u2

x + 1

2ρ2

). (3.9)

Substituting the identity (3.9) into ddt E(t), we obtain

dE(t)dt

= −∫

R

exuux(x, t)dx +∫

R

ex(uux)xx(x, t)dx

−∫

R

ex∂x

(u2 + 1

2u2

x + 1

2ρ2

)(x, t)dx

= ex ((uux)x − uux)

∣∣∣∞−∞− ex

(u2 + 1

2u2

x + 1

2ρ2

) ∣∣∣∞−∞

+∫

R

ex(

u2 + 1

2u2

x + 1

2ρ2

)(x, t)dx

=∫

R

ex(

u2 + 1

2u2

x + 1

2ρ2

)(x, t)dx,

where we used (3.6) and (3.7).Therefore, in the lifespan of the solution, we have that E(t) is an increasing

function with E(0) = 0, thus it follows that E(t) > 0 for t ∈ (0, T], i.e.,

E(t) =∫ t

0

∫R

ex(

u2 + 1

2u2

x + 1

2ρ2

)(x, τ )dxdτ > 0.

By the same argument, one can check that the following identity for F(t) istrue:

F(t) = −∫ t

0

∫R

e−x(

u2 + 1

2u2

x + 1

2ρ2

)(x, τ )dxdτ < 0.


In order to finish the proof, it is sufficient to let L(t) = 12 E(t) and l(t) =

12 F(t) respectively. �

It is really a very nice property for the two-component generalizedCamassa–Holm equation. No matter the profile of the compactly supportedinitial datum X0(x) is (no matter it is positive or negative), for any t > 0 inits lifespan, the solution u(x, t) is positive at infinity and negative at negativeinfinity. Moreover, the tail of the corresponding solution at infinity grows astime goes on, while the propagation speed for ρ(x, t) is finite.

Acknowledgements This paper is written under the guidance of Professor Yong Zhou. Someresults are motivated by Zhou’s works. Ni thanks to the referee for his/her careful reading andconstructive suggestions on the manuscript, which greatly improved the paper. This work ispartially supported by the Program for New Century Excellent Talents in Universities (GrantNo. NCET 07-0299), ZJNSF (Grant No. R6090109) and NSFC (Grant No. 10971197).

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An Operator Theoretic Interpretationof the Generalized Titchmarsh-Weyl Coefficientfor a Singular Sturm-Liouville Problem

Pavel Kurasov · Annemarie Luger

Received: 21 February 2008 / Accepted: 21 February 2011 / Published online: 12 March 2011© Springer Science+Business Media B.V. 2011

Abstract In this article an operator theoretic interpretation of the generalizedTitchmarsh-Weyl coefficient for the Hydrogen atom differential expressionis given. As a consequence we obtain a new expansion theorem in terms ofsingular generalized eigenfunctions.

Keywords Titchmarsh-Weyl coefficient · Singular differential operator ·Generalized Nevanlinna function · Supersingular perturbation

Mathematics Subject Classifications (2010) 30H15 · 34L40 · 47E05 · 81Q10

The authors gratefully acknowledge support from the Swedish Research Council, Grant#50092501, and the Austrian Science Fund (FWF), grant numbers P15540-N05 andJ2540-N13.

P. KurasovDepartment of Physics, St. Petersburg University,198904 St. Petersburg, Russia

P. Kurasov · A. Luger (B)Department of Mathematics, Stockholm University,106 91 Stockholm, Swedene-mail: [email protected]

P. Kurasove-mail: [email protected]

P. Kurasov · A. LugerDepartment of Mathematics, LTH, Box 118,221 00 Lund, Sweden

116 P. Kurasov, A. Luger

1 Introduction

The main object in this paper is the ordinary differential expression

�(y) := −y′′(x) +(

q0 + q1xx2

)y(x), x ∈ (0, ∞), (1)

with q0, q1 ∈ R, which is known as the ‘Hydrogen atom differential expression’(see [14], Section 39), since it appears after separation of variables in two-and three-dimensional Schrödinger equations with Coulomb potential. Thecorresponding differential equation

−y′′(x) +(

q0 + q1xx2

)y(x) = λy(x), (2)

is probably one of the most well studied equations in classical mathematicalphysics. Its solutions can be expressed in terms of Whittaker functions, or otherconfluent hypergeometric functions [1, 5]. However, we are going to make useof these special solutions only in the last part, starting with Section 4.1. Beforethat we actually only make use of asymptotic properties, since this approachwill be used also in upcoming work for more general potentials.

The differential expression (1) has two singular endpoints. It is in limit pointcase at ∞ (in the terminology of H. Weyl [3, 26]). Due to the non integrabilityof the potential at the origin also the left endpoint is singular. The mostimportant case for us is q0 � 3

4 , where limit point case prevails also at 0.Recall for a moment the case of a regular left endpoint. Then (usual)

the Titchmarsh-Weyl coefficient, which plays a crucial role in the spectralanalysis, is connected with the asymptotic behavior of the solutions to thedifferential equation (2). However, essentially the same function appears asKrein’s Q-function in the denominator of the resolvent formula, which givesall possible self-adjoint realizations of the differential expression, describedalso by boundary conditions at the origin. Summing up one associates withthe differential expression a Nevanlinna function which has a double nature: itappears both as Titchmarsh-Weyl coefficient and as Krein’s Q-function.

For the singular differential expression (1), however these two approachesdo not work directly. In Krein’s approach the operator family is reduced tojust one operator leaving no possibility for neither comparing resolvents ofdifferent operators nor imposing boundary conditions. Trying to follow theTitchmarsh-Weyl approach one finds that only one solution of (2) is regularat the origin. In [15, 16] it was suggested to overcome the latter problem byusing also the singular solution in order to define a generalized Titchmarsh-Weyl coefficient, which in this case appeared to be a generalized Nevanlinnafunction with degree of non-positivity estimated in terms of the parameter q0.

See also [17], where generalized Titchmarsh-Weyl coefficients are studied alsofor a wider class of potentials.

It is the aim of this paper to give an operator interpretation for thisgeneralized Titchmarsh-Weyl coefficient corresponding to the differential

On the Generalized Titchmarsh-Weyl Coefficient 117

expression (1). Constructing the model we aimed to satisfy the followingrequirements:

– The operators should be self-adjoint in a Hilbert space of functions and actas the differential expression (1).

– The family of self-adjoint operators should be given by a Krein-typeformula.

In order to meet these conditions we applied the theory of supersingularperturbations. In other words we obtain a family of self-adjoint operatorsacting in a new Hilbert space of (physically relevant but not necessarily square-integrable functions) and their domains are described by certain ‘boundaryconditions’. The main result, stated in Theorem 2, is that the generalizedNevanlinna function which describes this family coincides with the generalizedTitchmarsh-Weyl coefficient up to a polynomial. Its degree is bounded in termsof the parameters, which also implies that the number of negative squares of

the generalized Titchmarsh-Weyl coefficient equals[

1+√

14 +q0

2

].

This program has partially been carried out in the particular case of theBessel operator, i.e. when q1 = 0, in [12]. However, there a rather abstractPontryagin space model for the generalized Nevanlinna function, which isobtained by analytic continuation from the Q-function of the limit circle case(q0 < 3

4 ), is constructed.Finally we obtain a new kind of eigenfunction expansion involving not

square integrable functions which may be interpreted as scattered waves. Thisexpansion is proved using the model constructed in Section 3.

The paper is organized as follows. In the last part of this introduction,in Section 1.1, we shortly recall the situation for a Sturm-Liouville operatorwith one regular endpoint. Section 2 is devoted to the asymptotic behavior ofcertain solutions of (2).

In Section 3 the operator model is constructed explicitly. In Section 3.1 wefirst show that the perturbations we are interested in are indeed so-calledsupersingular perturbations and hence the corresponding theory can thenbe applied in Section 3.2. Section 3.3 contains the main result, which givesthe connection between the generalized Titchmarsh-Weyl-coefficient and thedenominator in the Krein-type formula describing the model.

Only in Section 4 we use classical results on the analytic behavior ofsolutions of (2). First we recall in Section 4.1 spectral properties of the classicalHydrogen atom operator and deduce from this corresponding results for themodel operators. Finally in Section 4.2 we find a new kind of expansionin terms of functions which are not square integrable at the origin. In theappendix, finally, basic facts on the scale of Hilbert spaces are collected anda short discussion of the limit circle case, i.e. − 1

4 < q0 < 34 , completes the

picture.Finally we want to mention that the Bessel operator (i.e. q1 = 0) has recently

also been investigated from completely different points of view. In [10] and [11]


approximation by regular differential operators and corresponding models arediscussed, and in [25] an indefinite canonical system is constructed.

1.1 Regular Case: Classical Theory

We briefly recall the situation in the case of a Sturm-Liouville-operatorcorresponding to the differential expression �(y) := −y′′ + qy on the half line[0, ∞), which is regular at 0, that is for the real potential q it holds q ∈L1

loc[0, ∞), and which is in limit point case at ∞. Under these assumptionsfor every λ ∈ C \ R the equation

�(y) = λy (3)

has exactly only one (up to a scalar multiple) solution which belongs to thespace L2(0, ∞). Hence with the basic solutions y1 and y2 of (3), which aredetermined by the Cauchy data

y1(0, λ) = 0 y2(0, λ) = 1

y′1(0, λ) = −1y′

2(0, λ) = 0,

the requirement

g(x, λ) := y2(x, λ) − m(λ)y1(x, λ) ∈ L2(0, ∞)

defines m(λ) uniquely. This function is usually called Titchmarsh-Weylcoef f icient of the differential expression �. It is a Nevanlinna function, m ∈ N0,that is, it is a symmetric function, i.e. m(λ) = m(λ), which maps the upperhalf plane C

+ holomorphically into itself. Its analytic properties are closelyconnected with the spectrum of the self-adjoint realizations of �. These realiza-tions, or in other words, self-adjoint extensions of the corresponding minimaloperator, which is defined on the domain C∞

0

((0, ∞)

), are given as restrictions

Lτ , τ ∈ R ∪ {∞}, of the differential expression � to the domain

dom(Lτ ) = {y ∈ L2(0, ∞), �(y) ∈ L2(0, ∞), y(0) − τy′(0) = 0}.

They are connected via

(Lτ − λ)−1 = (L0 − λ)−1 − 〈g(x, λ), · 〉m(λ) − 1

τ

g(x, λ), (4)

where L0 is the particular extension given by the Dirichlet boundary condition.Note that here and in the following we use the notation 〈 · , · 〉 for the innerproduct, such that it is linear in the second and conjugate linear in the firstargument.

On the other side the same differential expression can also be consideredusing methods of classical perturbation theory. Define the element ϕ :=(L0 − λ0)g( · , λ0), which in general does not belong to L2(0, ∞), but ratherϕ ∈ H−2(L0) since g ∈ L2(0, ∞). For more details on the rigged spaces H−n


see Appendix A below, cf. also [2, 4]. Then by standard techniques with thesingular perturbation

L0 + t〈ϕ, · 〉ϕ t ∈ R ∪ {∞}there is associated a whole family of self-adjoint operators in L2(0, ∞), whichare given by

(Lγ − λ)−1 = (L0 − λ)−1 − 〈(L0 − λ)−1ϕ, · 〉Q(λ) + γ

(L0 − λ)−1ϕ, γ ∈ R ∪ {∞},(5)

where in general the correspondence between t and γ is not fixed. Here Q is aQ-function corresponding to the symmetric operator S, which is defined as therestriction of L0 to those elements y for which 〈ϕ, y〉 = 0 and its self-adjointextension L0. Since g( · , λ) = (L0 − λ)ϕ the formulas (5) and (4) describethe same family of self-adjoint extensions. Moreover, Q(λ) − m(λ) is a realconstant.

In what follows we are giving a corresponding connection for the singulardifferential expression (1).

2 Asymptotic Analysis and the Generalized Titchmarsh-Weyl Coefficient

In this section we study the solutions of the differential equation

−y′′(x) +(q0

x2+ q1

x

)y(x) = λy(x) x ∈ (0, ∞), λ ∈ C, (6)

where q1 ∈ R and q0 > − 14 , with respect to their asymptotic behavior at the

singular endpoints and introduce the generalized Titchmarsh-Weyl-coefficient.We want to point out that here we are making use of the asymptotic propertiesof equation (6) only, rather than using its explicit solutions.

2.1 Asymptotics at the Origin

We follow the lines of [15] but also extend the analysis there. Note that theequation (6) is of Fuchsian type with a weak singularity at the point x = 0.Hence Frobenius theory can be applied and the solutions can be obtained viathe generalized power series Ansatz

y(x, λ) = xα

∞∑j=0

a j(λ)x j with a0 = 0. (7)

Then the corresponding index equation turns out to be

α2 − α − q0 = 0

which has the real solutions α± := 12 ±

√14 +q0, with α−<α+ and α−+α+ = 1.

Two particular solutions g+(x, λ) and g−(x, λ) (the so-called ‘regular’ and


‘singular’ solutions) corresponding to the indices α+ and α−, respectively,will play an important role. The following lemma summarizes some of theirasymptotic properties. Here and in the following [x] denotes the integer partof x.

Lemma 1 Set α± = 12 ±

√14 + q0. Then equation (6) has two linearly indepen-

dent solutions satisfying

g+(x, λ) =∞∑j=0

a j(λ)xα++ j

g−(x, λ) =m0−1∑

j=0

c j(λ)xα−+ j + o(xα+−1

)for x → 0+ (8)

with m0 := [α+ − α−] and coef f icients a j and c j given by the recursion

t j+2 = q1t j+1 − λt j

( j + 2)(2α + j + 1)(9)

with α = α+ and starting values a0 = 1, a1 = q1

2α+ and with α = α− and starting

values c0 = 1α−−α+ , c1 = q1

2α− c0, respectively. Moreover the following holds:

(i) The functions g+(x, λ) and g−(x, λ) and their derivatives with respect to xare entire in λ for every x ∈ (0, ∞) and

g±( · , λ) = g±( · , λ) and g′±( · , λ) = g′±( · , λ).

(ii) With the notation W for the Wronskian

W(y1(x), y2(x)

) := y1(x)y′2(x) − y′

1(x)y2(x)

it holds for all λ, z ∈ C:

W(g+(x, λ), g−(x, λ)

) ≡ 1

limx→0

W(g+(x, λ), g−(x, z)

) = 1

limx→0

W(g+(x, λ), g+(x, z)

) = 0

limx→0

W(g−(x, λ), g−(x, z)

) ={

0 if q0 < 34

∞ if q0 � 34

.

Note that with obvious adjustments the lemma also holds for q0 = − 14 .

Throughout the whole paper we are going to write limx→0

instead of limx→0+

,

hoping that this will not lead to any missunderstanding since we always havex > 0.

Proof From (7) recursion (9) and hence also the expansion of g+ followdirectly. The classical theory shows that in order to obtain a second linear


independent solution of (6) one has to distinguish two different cases. Ifα+ − α− ∈ N, then a ‘singular’ solution is of the form

g−(x, λ) :=∞∑j=0

c j(λ)xα−+ j with c0 = 1

α− − α+, (10)

where c j satisfy the recursion (9) with α = α−. Here the choice of c0 is donesuch that the Wronskian in (ii) is normalized. If, however, α+ − α− = m0 ∈N \ {0} then the second solution is obtained by the Ansatz

g−(x, λ) :=∞∑j=0

d j(λ)xα−+ j + K(λ) ln x g+(x, λ), (11)

where for normalization reasons we again choose d0 = 1α−−α+ , which immedi-

ately implies d1 = q1

2α− d0. Then for 0 � j < m0 − 2 the Ansatz yields

d j+2 = q1d j+1 − λd j

( j + 2)(2α− + j + 1).

The coefficient dm0 appears to be arbitrary, however, its choice does not effect(8). The corresponding equation yields

K(λ) = dm0−1q1 − dm0−2

m0,

where we set d−1 = 0 in case m0 = 1. For sake of completeness we also addthat for j > m0 − 2 the Ansatz (11) gives

d j+2 = − q1d j+1 − λd j

( j + 2)(m0 − 2 − j)+ K(λ)a j+2−m0

2 j + 4 − m0

( j + 2)(m0 − 2 − j).

Summed up, this shows that in both cases the claimed expansion for g−holds. The other statements follow then directly from the asymptotic expan-sions (8). For more details see also [15]. �

Note that in the above proof in the special case α+ − α− ∈ N there was norequirement on the coefficient dm0 since then α− + m0 = α+. However, it turnsout to be possible to choose dm0 such that the following further refinement ofexpansion (8) for g− holds.

Corollary 1 Equation (6) has a solution g− which is of the form

g−(x, λ) =m0∑j=0

c j(λ)xα−+ j + K(λ)xα+ ln x

+ xm0+α−+1(H1(x, λ) + ln x H2(x, λ)

), (12)

where m0 = [α+ − α−], the coef f icients c j(λ) and K(λ) are polynomials in λ

of degree �[m0

2

], and the functions H1 and H2 are both entire in λ and

holomorphic at x = 0.


Proof We use the notations from the proof of the previous lemma. If α+ −α− ∈ N then (10) gives directly K(λ) = 0 and H2(x) = 0 and, indeed recursion(9) implies that c j(λ) is a polynomial in λ of degree

[ j2

]. If α+ − α− ∈ N, then

(12) follows immediately from (11), and here only dm0(λ) has to be chosen as apolynomial in λ with degree �

[m02

]. �

Remark 1 Let q0 > − 14 . From Lemma 1 one sees that for all λ ∈ C and every

fixed x0 ∈ R+ it holds

g+( · , λ) ∈ L2(0, x0),

g−( · , λ) ∈ L2(0, x0) if and only if − 14 < q0 < 3

4 ,.

Hence for the differential expression � prevails limit point case at the singularendpoint 0 if and only if q0 � 3

4 .

2.2 Asymptotics at ∞

The endpoint ∞ is—under our assumptions—always in limit point case. Thusfor λ ∈ C \ R there is (up to a constant factor) exactly one linear combinationof g+ and g− which is square integrable in a neighborhood of ∞:

g( · , λ) := g−( · , λ) − m(λ)g+( · , λ) ∈ L2(x0, ∞) for x0 ∈ R. (13)

The function m, defined by (13), is called generalized Titchmarsh-Weylcoef f icient. For the differential expression under consideration it has beenintroduced in [15] and further investigated in [16], see also [17]. Note that byclassical arguments this definition can be extended also to λ < 0 except theeigenvalues of the Hydrogen atom operator.

Actually, m can be calculated even explicitly, see Section 4.1, where alsofurther properties of the function g are deduced. However, for this moreaccurate analysis we are going to make use of the explicit form of the solutionsof (6), which are not needed in the first part of this paper.

Remark 2 Note that the so defined function m heavily depends on the par-ticular choice of g± as basic solutions, even if this basis seems to be naturalfor the problem. However, let us mention that for potentials where Frobeniustheory is not available the choice of the basic solutions becomes a crucialquestion.

2.3 Regularizations

Locally at 0 the function g in (13) behaves as the singular solution g−, thatis, g(x, λ) = O(xα−) as x → 0+. However, the first two coefficients c0 andc1 in expansion (8) actually do not depend on the spectral parameter. Thusthe difference g(x, μ1) − g(x, μ2) = O(xα−+2) as x → 0+, and hence is less


singular at the origin. This gives rise to the following definition of higher orderdifferences.

Here and in the following we use the notations

R− = (−∞, 0), R

−0 = (−∞, 0],

and for R+ and R

+0 accordingly. Let μ1, . . . , μk ∈ R

− not be eigenvalues of thehydrogen atom operator and mutually different for k � 1, then define

gk(x) :=k∑

i=1

A(k)

i g(x, μi) (14)

with coefficients A(k)

i for 1 � i � k such that A(1)1 := 1 and for k > 1

A(k)

i := − 1

μk − μiA(k−1)

i for i < k and A(k)

k := −k−1∑i=1

A(k)

i . (15)

Remark 3 Note that it holds (� − μk)gk = gk−1 for k > 1, and (� − μ1)g1 = 0,where � denotes the differential expression (6).

The next lemma shows that the regularity of these functions indeed in-creases with k. In Theorem 1 we will later also give an operator theoreticexplanation for this fact. Let us first introduce the number

n := 2 +[√1

4+ q0

],

which will play an important role in the following.

Lemma 2 Assume q0 � 34 , let k � n − 2 = [√ 1

4 + q0], and m0 = [α+ − α−].

Then the functions gk(x) (def ined in (14)) have the asymptotic expansions

gk(x) =m0−1∑

j=2(k−1)

C(k)

j xα−+ j + o(xα+−1) as x → 0+ (16)

with some C(k)

j ∈ R, where the f irst coef f icient C(k)

2(k−1)= 0.

Proof Note first that under the above assumptions the sum in (16) is notempty. Using (8) we can write

gk(x) =m0−1∑

j=0

C(k)

j xα−+ j + o(xα+−1) as x → 0+


with C(k)

j :=∑ki=1 A(k)

i c j(μi), where the coefficients A(k)

i are given in (15) and

c j as in Lemma 1. We have to show that C(k)

j = 0 for j < 2(k − 1). For k = 1the above statement is already included in Lemma 1. Hence let us now assumek > 1. Since c0(λ) and c1(λ) do not depend on λ we have directly

C(k)

j = c j

k∑i=1

A(k)

i = 0 for j = 0, 1 and k > 1. (17)

The defining recursion (15) for the A(k)

i gives

C(k)

j =k−1∑i=1

A(k−1)

i

μk − μi

(c j(μk) − c j(μi)

). (18)

By using recursion (9) for the c j for j > 1 and then again (18) we get

C(k)

j = 1

j(2α− + j − 1)

(q1C(k)

j−1 − μkC(k)

j−2 − C(k−1)

j−2

).

Going on like this one obtains in finitely many steps that C(k)

j is a linear

combination of C(i)0 and C(i)

1 with i � k. Here i > 1 as long as j < 2(k − 1)

and hence C(k)

j equals zero by (17). However, for j = 2(k − 1) this implies

furthermore C(k)

2(k−1)= 0. �

Remark 4 For the first non-vanishing coefficients the following recursionrelation holds

C(k)

2(k−1)= − C(k−1)

2(k−2)

2(k − 1)(2α− + 2k − 3).

3 Perturbations in the Limit Point Case

3.1 Recognizing the Perturbation as Supersingular

In what follows we concentrate on the case q0 � 34 only, that is � is in limit point

case at 0, or in other words, the maximal operator is self-adjoint:

dom L0 := {y ∈ L2(0, ∞)|y, y′ ∈ ACloc(0, ∞), �(y) ∈ L2(0, ∞)} (19)

and

L0 y := �(y).

Remark 5 Since in this case every y ∈ dom Lmax satisfies the boundary con-dition limx→0 W(y(x), g+(x, λ0)) = 0 the above notation is in accordance withthe one used in the limit circle case, see also Appendix A.


Motivated from the regular case we are interested in perturbations formallygiven by

L0 + t〈ϕ, · 〉ϕ t ∈ R ∪ {∞}

with ϕ = (L0 − λ0)g( · , λ0), where g( · , λ0) is the function given in (13). How-ever, here g( · , λ0) is not square integrable locally at 0, and hence we need somemore considerations in order to make the definition of ϕ precise and identifyit as an element from H−n(L0) for some n ∈ N. As a first step the next lemmagives estimates for functions f ∈ dom Lk

0 .

Lemma 3 For f ∈ L2(0, ∞) the resolvent of L0 is given by

((L0 − λ)−1 f

)(x) = −g(x, λ)

∫ x

0g+(s, λ) f (s) ds

−g+(x, λ)

∫ ∞

xg(s, λ) f (s) ds. (20)

Let the integer k satisfy k �[

3+√

14 +q0

2

], then for every f ∈ dom Lk

0 there exists

a constant C > 0 such that for some f ixed x0 > 0 it holds for all x ∈ (0, x0):

| f (x)| � Cx− 12 +2k and | f ′(x)| � Cx− 3

2 +2k. (21)

Proof For λ in the resolvent set ρ(L0) denote by R(λ) f the integral on theright side of (20), which is well defined for every f ∈ L2(0, ∞). We first proveequality (20) for f ∈C∞

0

((0, ∞)

). In particular, we have to show that R(λ) f is

square integrable locally at 0 and ∞, provided that f has compact supportin the interval (0, ∞). We choose positive real numbers a and b such thatsupp f ⊂ (a, b). Note that

(R(λ) f

)(x) = −g+(x, λ)

∫supp f

g(s, λ) f (s) ds for x < a

and

(R(λ) f

)(x) = −g(x, λ)

∫supp f

g+(s, λ) f (s) ds for x > b .

This also shows R(λ) f∣∣(0,a)

∈ L2(0, a) and R(λ) f |(b ,∞) ∈ L2(b , ∞). UsingLemma 1 it is straight forward to see �(R(λ) f ) = λR(λ) f and hence R(λ) f =(L0 − λ)−1 f for every f ∈C∞

0

((0, ∞)

).


For each f ∈ L2(0, ∞) there exists then a sequence fn ∈ C∞0

((0, ∞)

)such

that ‖ f − fn‖L2 → 0. Since the resolvent (L0 − λ)−1 is a bounded operator itholds

(L0 − λ)−1 f = L2- limn→∞(L0 − λ)−1 fn = L2- lim

n→∞ R(λ) fn.

For every x ∈ (0, ∞), however, the continuous functions(R(λ) fn

)(x) converge

to the continuous function(R(λ) f

)(x), and hence this pointwise limit coincides

with the L2-limit, which finally gives

(L0 − λ)−1 f = R(λ) f for every f ∈ L2(0, ∞).

In order to show (21) we use mathematical induction. The asymptotic expan-sion (8) for g+ and Cauchy-Schwarz-inequality imply for every x ∈ (0, x0) withsome fixed x0

|g(x, λ)

∫ x

0g+(s, λ) f (s) ds| � C1xα−

∫ x

0sα+| f (s)| ds

� C2xα−+α++ 12 = C2x

32 (22)

and

|g+(s, λ)

∫ ∞

xg(s, λ) f (s) ds| � C3xα+

( ∫ x0

xsα−| f (s)| ds +

∫ ∞

x0

|g(s, λ) f (s)| ds)

� C4xα++α−+ 12 + C5xα+ � C6x

32 , (23)

where we have used that if q0 � 34 then α+ � 3

2 . If f ∈ dom Lk+10 , that is

f = (L0 − λ)−1h with some h ∈ dom Lk0 and λ ∈ R

−. Then using (21) for h oneobtains the corresponding estimate as in (22) and noting − 1

2 + 2k � α+ also(23). This proves the first estimate in (21) for k + 1. In the same way the secondestimate can be shown. �

The following theorem establishes the connection between the operatorL0 and the functions g( · , λ) and gk defined in (13) and (14), respectively. Inparticular, it makes the definition of ϕ := (L0 − λ0)g( · , λ0) precise.

Theorem 1 Let g and gk be given as above and n = 2 +[√

14 + q0

]. Then the

element ϕ := (L0 − λ0)g( · , λ0) is independent of the particular choice of λ0 ∈R

− with L0 − λ0 > 0 and

ϕ ∈ H−n(L0) \ H−n+1(L0).

Furthermore, it holds

gk = (L0 − μk)−1 . . . (L0 − μ1)

−1ϕ (24)

and, in particular,

gk ∈ H−n+2k(L0) \ H−n+2k+1(L0).


Remark 6 In case q1 = 0 and α+ − α− ∈ Neven a modified Hankel transformwas applied to the problem in [12], and then these statements become obvious.However, this transformation makes essential use of well known propertiesof Bessel functions and hence in the general case we prove the theoremdifferently.

Proof Since we consider the scale of Hilbert spaces corresponding only tothe operator L0, within this proof we are going to write simply Hs instead ofHs(L0). Lemma 2 implies that for some large enough index m the function gm

belongs to H0 \ H2. In order to determine the number m note that the latteris equivalent to gm ∈ L2(0, ∞) but �(gm) ∈ L2(0, ∞). Since �(gm) = μmgm +gm−1 the asymptotic expansion (16) implies that this is further equivalent to

2(α− + 2(m − 1)

)> 1 ∧ 2

(α− + 2(m − 1) − 2

)� −1,

from which we can conclude

m =⎡⎣3 +

√14 + q0

2

⎤⎦ . (25)

In the next step we show that for k = 1, . . . , m it holds

gk ∈ H−2(m−k) \ H−2(m−k)+2 (26)

and

(L0 − μk)gk = gk−1. (27)

Consider first k = m. Then we already have gm ∈ H0 \ H2. Hence (L0 − μm)gm

is an element from H−2 and we are going to show that, in fact, it coincideswith the function gm−1. To this end apply (L0 − μm)gm to an arbitrary f ∈ H2,that is

〈 f, (L0 − μm)gm〉 =∫ ∞

0gm(x)

((L0 − μm) f

)(x) dx

= limε→0

∫ ∞

ε

gm(x)(� − μm) f (x) dx.

Integrating by parts leads to

limε→0

W(

f (ε), gm(ε))+

∫ ∞

ε

(� − μm)gm(x) f (x) dx.

Here the first limit exists according to Lemma 3 and equals 0. Using againRemark 3 finally gives

〈 f, (L0 − μm)gm〉 =∫ ∞

0gm−1(x) f (x) dx,

which is (27) for k = m. Next we reduce the number k step by step. Assumethat the relations (26) and (27) already hold for some k > 1. Then (27), in


particular, implies gk−1 ∈ H−2(m−k)−2 \ H−2(m−k). Take now an arbitrary func-tion f ∈ H2(m−k)+2 = dom Lm−k+1

0 and consider 〈 f, (L0 − μk−1)gk−1〉 as above.Then (27) for k − 1 follows in the same way, where again the estimates (21)are essential and (24) is proved for k � m. We leave the details to the reader.

Since we know now that, in particular, g( · , μ1) ∈ H−2m+2 \ H−2m+4 theelement ϕ := (L0 − μ1)g( · , μ1) is well defined and belongs to H−n for n either2m − 1 or 2m. In the next step of the proof we are going to determine nprecisely. Obviously ϕ ∈ H−2m+1 \ H−2m+2 exactly if gm ∈ H1 \ H2, that is, gm

belongs to the domain of the quadratic form associated with the operator L0

but not to the operator’s domain, or in other words, this happens if and only ifthe following integral converges:∫ ∞

0

(|g′

m(x)|2 + q0 + q1xx2

|gm(x)|2)

dx,

but the integral∞∫0

|�(gm(x))|2 dx diverges. By integration by parts the quadratic

form becomes

limε→0

−g′m(ε)gm(ε) +

∫ ∞

ε

�(gm(x))gm(x) dx. (28)

From (16) it follows that both the boundary term and the integral term havean expansion starting with ε2(α−+2(m−1))−1. The only exception here is if 2(α− +2(m − 1)) − 1 = 0, then the integral starts with a logarithmic term. Hence if2(α− + 2(m − 1)) − 1 > 0 the limit (28) exists, if however 2(α− + 2(m − 1)) −1 < 0 we have to investigate the leading coefficient, which equals

−(α− + 2(m − 1))(

C(m)

2(m−1)

)2 − C(m−1)

2(m−2) C(m)

2(m−1)

2(α− + 2(m − 1)) − 1,

Inserting the recursions for the coefficients from Remark 4 this further equals(C(m)

2(m−1)

)2−(α−+2(m−1))(2(α−+2(m−1))−1)+2(m−1)(2α−+2m−3)

2(α−+2(m−1))−1 . The numerator can fur-ther be simplified to

−(α− + 2(m − 1))2 − α−(α− − 1). (29)

Since in this section we assumed limit circle case α−(α− − 1) = q0 > 0 and thus(29) can not vanish. Hence the limit in (28), indeed, exists if and only if theinequality 2(α− + 2(m − 1)) − 1 > 0 holds. Inserting the formula (25) for m

one easily finds that this inequality is satisfied if and only if[√

14 + q0

]is an

odd number. In this case ϕ ∈ H−2m+1 \ H−2m+2 and 2m − 1 can be written as

2m − 1 = [√ 14 + q0

]+ 2. In the other case, however, ϕ ∈ H−2m \ H−2m+1 and

then 2m = [√ 14 + q0

]+ 2. Hence in both cases it holds

ϕ ∈ H−n \ H−n+1 for n =[√1

4+ q0

]+ 2.


We show now that ϕ := (L0 − μ1)g( · , μ1) is independent of the particularchoice of μ1. To this end apply ϕ to f ∈ Hn. Integration by parts gives

〈ϕ, f 〉 = 〈g( · , μ1), (L0 − μ1) f 〉 =∫ ∞

0g(x, μ1)(� − μ1) f (x) dx

= limε→0

W(g(ε, μ1), f (ε)).

Since the asymptotic expansions

g(ε, μ1) = 1

α− − α+εα− + O(εα−+1), g′(ε, μ1) = α−

α− − α+εα−−1 + O(εα−)

hold for ε → 0, Lemma 3 implies

limε→0

W(g(ε, μ1), f (ε)) = 1

α− − α+limε→0

W(εα− , f (ε)),

which indeed is independent of the point μ1. Finally (27), or equivalently

(L0 − μk)−1gk−1 = gk

follows also for k > m directly by using the defining relation (14) for thefunctions gk−1 and applying the resolvent equation to the expression (L0 −μk)

−1(L0 − μi)−1ϕ. �

Remark 7 Recall that the elements gi are actually usual functions, but notnecessarily square integrable locally at 0. However, the element ϕ is a singulardistribution with support at the point x = 0 only:

〈ϕ, f 〉 = 1

α− − α+limx→0

W(xα− , f (x)) for f ∈ Hn(L0).

Theorem 1 enables us now to give a meaning to the formal expression L0 +t〈ϕ, · 〉ϕ by using the concept of supersingular perturbations.

3.2 Operator Model

Consider the formal expression

L0 + t〈ϕ, · 〉ϕ, t ∈ R, (30)

where L0 is a self-adjoint semi-bounded linear operator acting in a Hilbertspace H and ϕ ∈ H−n(L0) \ H−n+1(L0) is a singular element. We are going todescribe a family of model operators developed in the series of papers [20, 21,23, 24] and modified in [6, 22]. We mention that an alternative approach usingPontryagin spaces was carried out in [8, 28, 30].

Motivated by the regular situation one might be intended to consider(non-trivial) self-adjoint extensions of the symmetric restriction L0 |{ψ :〈ϕ,ψ〉=0}.


However, in case n � 3, when the perturbation is called supersingular, observethe following two facts:

– The restriction L0 |{ψ∈dom (L0):〈ϕ,ψ〉=0} is essentially self-adjoint in H (andhence has only trivial self-adjoint extensions). However, considered as anoperator in the Hilbert space Hn−2(L0) with domain in Hn(L0) it becomessymmetric with defect (1, 1).

– Since the Krein type formula for the resolvents should be kept, elements ofthe form (L0 − μ)−1ϕ ∈ H−n+2(L0) \ H−n+3(L0), which do not even belongto the space H0, have to be included.

These requirements lead to model operators acting in the restricted ex-tended space

H := Cn−2 ⊕ Hn−2(L0). (31)

Every element U := (u, U) ∈ H can be identified with an element fromH−n+2(L0) by the following natural embedding

ρU :=n−2∑j=1

u jg j + U, (32)

where again the notation g j = (L0 − μ j)−1 . . . (L0 − μ1)

−1ϕ is used with dis-tinct points μi ∈ R ∩ �(L0). The vector space H is equipped with the scalarproduct

〈U, V〉H := 〈u, �v 〉Cn−2 + 〈U, b n−2(L0)V〉H0 , (33)

where U := (u, U) and V := (v, V), with u, v ∈ Cn−2, and the functions U, V ∈

Hn−2(L0). Here � = �∗ is a Gram matrix, and b n−2 denotes the regularizingpolynomial, which is convenient to chose as

b n−2(λ) := (λ − μ1)(λ − μ2)...(λ − μn−2). (34)

Note that the norm given by the inner product 〈U, b n−2(L0)V〉H0 is equiv-alent to the standard norm in the space Hn−2(L0) corresponding to 〈U, (L0 −μ1)

n−2V〉H0 . In what follows we skip the index 〈 · , · 〉H0 and simply write 〈 · , · 〉.Let M denote the (n − 2) × (n − 2) matrix

M :=

⎛⎜⎜⎜⎜⎜⎜⎜⎝

μ1 1 0 . . . 0 00 μ2 1 . . . 0 00 0 μ3 . . . 0 0

. . . . . . . . .. . . . . . . . .

0 0 0 . . . μn−3 10 0 0 . . . 0 μn−2

⎞⎟⎟⎟⎟⎟⎟⎟⎠

.

In the following we assume that the Gram matrix � is positive definite andsatisfies

�M − M∗� = 0, (35)


i.e. the matrix M is Hermitian with respect to the scalar product given by theGram matrix �. It has been shown in [6] that such a choice is possible exactlyif the regularization points μi are mutually distinct.

Under these conditions the following proposition, which was proven in [6],describes the family of self-adjoint model operators.

Proposition 1 Let θ ∈ [0, π) and en−2 := (0, ..., 0, 1). Then the operator Lθ

def ined on the domain

dom (Lθ ) :={

U = (u, U) ∈ H :U = un−1gn−1 + Ur, un−1 ∈ C, Ur ∈ Hn(L0),

cos θ un−1 + sin θ (〈ϕ, Ur〉 − 〈en−2, �u〉Cn−2) = 0}

(36)

acting as

Lθ

(uU

):=(

Mu + un−1en−2

L0Ur + μn−1un−1gn−1

)

is self-adjoint in H, provided that � satisf ies (35).

Remark 8 Note that—up to the embedding �—such an operator acts as thedifferential expression �, that is,

��U = �LθU,

restricted to certain elements satisfying the generalized ‘boundary condition’(36).

It is a straight forward calculation to see that Krein’s formula here takes thefollowing form

(Lθ − λ)−1 = (L0 − λ)−1 −⟨�(λ), · ⟩

H

Q(λ) + cot θ�(λ), (37)

where L0 is the operator corresponding to θ = 0, that is, L0 = M ⊕ L0, thevector

�(λ) :=(

(M − λ)−1en−2

(L0 − λ)−1gn−2

)∈ H (38)

denotes the corresponding defect element and the Q-function takes the form

Q(λ) := (λ − μn−1) 〈�(μn−1), �(λ)〉H

+ ⟨en−2, �(M − μn−1)

−1en−2⟩Cn−2 . (39)

From (37) also the restricted-embedded resolvent �(Lθ − λ)−1|Hn−2(L0) in theform of Krein’s formula can be deduced.


Proposition 2 Let the operators L0 and Lθ and the function Q(λ) be given asabove, and the natural embedding � : H → H−n+2(L0), def ined in (32). Then itholds

ρ(Lθ − λ)−1|Hn−2(L0) = (L0 − λ)−1 − 1

b n−2(λ)(Q(λ) + cot θ)

×〈(L0 − λ)−1ϕ, · 〉(L0 − λ)−1ϕ, (40)

where the polynomial b n−2 is def ined in (34).

The essential step in order to proof this theorem is to verify the embeddingof the defect element

�� = 1

b n−2(λ)(L0 − λ)−1ϕ. (41)

Remark 9 Note that in the resolvent formula (40) the function

d(λ) := b n−2(λ)(Q(λ) + cot θ) (42)

is a generalized Nevanlinna function, since Q as the Q-function for operatorsin a Hilbert space is a usual Nevanlinna function.

3.3 Titchmarsh-Weyl Coefficient and Q-function

One of the main results of this paper is the following link between thegeneralized Titchmarsh-Weyl coefficient m (in the analytic approach) and d(in the above singular perturbation approach).

Theorem 2 Let d be the generalized Nevanlinna function in (42) and m be thegeneralized Titchmarsh-Weyl-coef f icient in (13). Then it holds

d(λ) − m(λ) = δ(λ),

where δ is a polynomial of degree less or equal to n − 2 =[√

14 + q0

].

Proof Expanding (39) we find that d can be written as

d(λ) = b n−2(λ)QL0(λ) + pn−2(λ), (43)


where QL0(λ) := (λ − μn−1)〈ϕ, (L0 − λ)−1gn−1〉 and pn−2 denotes a polyno-mial of degree � n − 2. By integrating by parts the first summand in (43) canbe rewritten as

b n−2(λ)〈(λ − μn−1)g( · , λ), gn−1( · )〉

= limε→0

b n−2(λ)

∫ ∞

ε

gn−1(x)(� − μn−1 − (� − λ)

)g(x, λ) dx

= limε→0

(λ − μ1) . . . (λ − μn−2)[(− gn−1( · )g′( · , λ) + g′

n−1( · )g( · , λ))∣∣∣∞

ε

+∫ ∞

ε

gn−2(x)g(x, λ) dx],

and repeating this calculation with each factor (λ − μi) leads to

= limε→0

W(G(ε, λ), g(ε, λ)

)(44)

with G( · , λ) :=n−1∑k=1

b k−1(λ)gk( · ). Here b k−1(λ) :=k−1∏j=1

(λ − μ j) and b 0(λ) := 1.

According to (14) the function G can be written as

G( · , λ) =n−1∑k=1

k∑i=1

b k−1(λ)A(k)

i g( · , μi)

where the coefficients A(k)

i were defined in (15). Expanding (44) by using (13)gives

limε→0

W(G(ε, λ), g(ε, λ)

)

= limε→0

W

(n−1∑k=1

k∑i=1

b k−1(λ)A(k)

i g−(ε, μi), g−(ε, λ)

)(45)

−n−1∑k=1

k∑i=1

m(μi) b k−1(λ)A(k)

i limε→0

W(g+(ε, μi), g−(ε, λ)

)(46)

− m(λ)

n−1∑k=1

k∑i=1

b k−1(λ)A(k)

i limε→0

W(g−(ε, μi), g+(ε, λ)

)(47)

+ m(λ)

n−1∑k=1

k∑i=1

m(μi) b k−1(λ)A(k)

i limε→0

W(g+(ε, μi), g+(ε, λ)

)(48)


According to Lemma 1 the limits in (46), (47), and (48) are 1, −1, and 0,respectively. Note that the remaining factor in (47) becomes

n−1∑k=1

b k−1(λ)

k∑i=1

A(k)

i = b 0(λ)A(1)1 +

n−1∑k=2

b k−1(λ)

k∑i=1

A(k)

i = 1.

In order to see that the limit in (45) vanishes we have a closer look at theasymptotic behavior of the function

∑n−1k=1

∑ki=1 b k−1(λ)A(k)

i g−(ε, μi). In factwe will show that the relevant terms in the expansion coincide with those ofg−(ε, λ). To this end we need the following technical lemma, which will beshown just after the current proof.

Lemma 4 Let the polynomials b k−1(λ) and the coef f icients A(k)

i be given asabove. Then for all l � n − 2 it holds

n−1∑k=1

k∑i=1

b k−1(λ)A(k)

i μli = λl.

Since in expansion (12) of the function g−( · , λ) the coefficients c j(λ) forj = 0, . . . , m0 and K(λ) are polynomials of degree �

[m02

] = n − 2 the abovelemma implies

W(n−1∑

k=1

k∑i=1

b k−1(λ)A(k)

i g−(ε, μi), g−(ε, λ)

)

= W( m0∑

j=1

c j(λ)εα−+ j+K(λ)εα+ +εm0+α−+1(h1(ε, λ)+ln x h2(ε, λ)),

m0∑j=1

c j(λ)εα−+ j+K(λ)εα+ +εm0+α−+1(h3(ε, λ)+ln x h4(ε, λ))

).

where hi for i = 1 . . . , 4 are holomorphic at ε = 0. Since the singular terms ofthe two functions here coincide the limit for ε → 0 is zero, and we finally obtainfrom (44)

d(λ) − m(λ) = pn−2(λ) −n−1∑k=1

b k−1(λ)

k∑i=1

m(μi)A(k)

i ,

which, indeed, is a polynomial of degree � n − 2. �

Proof of Lemma 4 We note first that the coefficients A(k)

i , which weredefined recursively in (15), can also be given explicitly by derivatives b ′

k ofthe polynomials b k

A(k)

i = 1

b ′k(μi)

for i = 1, . . . , k. (49)


Indeed, it is easy to see that the sequence in (49) satisfies the recursion in

(15) for i � k − 1. The remaining equality A(k)

k = −k−1∑i=1

A(k)

i can be seen by

multiplying the identity

1

b k(λ)=

k∑i=1

1

b ′k(μi)

1

λ − μi

with λ and then taking the limit λ → ∞. Hence in order to complete the proofwe have to show

n−1∑k=1

b k−1(λ)

k∑i=1

μli

b ′k(μi)

= λl for l = 0, 1, . . . , n − 2. (50)

For k � l + 1 the partial fractional decomposition

λl

b k(λ)=

k∑i=1

μli

b ′k(μi)

1

λ − μi

implies

k∑i=1

μli

b ′k(μi)

={

0 k > l + 11 k = l + 1

. (51)

Hence the polynomial

Pl(λ) := −λl +n−1∑k=1

b k−1(λ)

k∑i=1

μli

b ′k(μi)

(52)

is of degree � l − 1. We are calculating its values at the points μ1, . . . , μl.Changing the order of summation in (52) and taking into account (51) weimmediately obtain Pl(μ1) = 0 and

Pl(μ j) =j−1∑i=1

μli

j∑k=i

b k−1(μ j)

b ′k(μi)

for j = 2, . . . , l. (53)

Changing the indices to j =: i + N − 1 with i � 1 and N � 2 and introducingλm+1 := μi+m the inner sums in (53) become

j∑k=i

b k−1(μ j)

b ′k(μi)

= bi−1(λN)

b ′i(λ1)

· Ik1 ,

where INk := 1 +

N−k−1∑m=1

m∏m1=1

λN − λm1

λ1 − λm1+1−

N−k−1∏m2=1

λN − λm2+1

λ1 − λm2+1. It is easy to see

that INk = Ik+1 for k � N − 3 and hence

IN1 = IN

N−2 = 1 + λN − λ1

λ1 − λ2− λN − λ2

λ1 − λ2= 0.


Thus the summands in (53) vanish and the polynomial Pl with degree at mostl − 1 has l distinct zeros, and hence vanishes identically, which finally provesrelation (50). �

In [16] it was shown that the generalized Titchmarsh-Weyl coefficient m is ageneralized Nevanlinna function with negative index κ = [ n−1

2

]. As a corollary

of Theorem 2 we obtain an independent proof for this fact.

Corollary 2 The generalized Titchmarsh-Weyl-coef f icient m, introduced in(13), belongs to the generalized Nevanlinna class Nκ where

κ =[n − 1

2

]=⎡⎣1 +

√14 + q0

2

⎤⎦ .

Proof In [6, Sections 4.4 and 5.2] it was shown that the function d admits alsoa minimal representation in a certain Pontryagin space and hence belongs tothe class Nκ with κ = [ n−1

2

]. In fact, it belongs even to the class N∞

κ , see [7, 9],and hence it has an irreducible representation of the form

d(λ) = (λ2 + 1)κd0(λ) + p2κ−1(λ), (54)

where d0 is a usual Nevanlinna function satisfying

limy→∞

Im d0(iy)

y= 0, lim

y→∞ y Im d0(iy) = ∞

and p2κ−1 is a polynomial of degree � 2κ − 1. According to Theorem 2 thedifference between m and d is a polynomial of degree at most n − 2 � 2κ .Hence the function m admits an irreducible representation of the form (54)as well, which, in particular, implies m ∈ Nκ . �

4 Spectral Analysis

4.1 Spectral Properties

Here we are collecting some well known spectral properties of the classicalHydrogen atom operator L0 and derive corresponding properties for themodel operators Lθ . To this end recall the following facts for Whittaker-functions, see [31, Sections 16.1–16.4].

Let Wl,m(z) be the Whittaker function of index (l, m), which is well defined(as a contour integral) for all values l, m ∈ C. It is analytic for z ∈ C \ (−∞, 0]and satisfies in the domain |argz| < π

Wl,m(z) = e− z2 zl(

1 + O(

1

|z|))

(55)


as z → ∞. The functions Wl,m(z) and W−l,m(−z) form a fundamental systemof solutions of the so-called Whittaker-equation:

d2

dz2W(z) +

(−1

4+ l

z+

14 − m2

z2

)W(z) = 0. (56)

The following result is well known (see eg. [29, Section 4.17], [14]), we aregoing to give only a sketch of the proof, as far as we will make use of thearguments later on.

Proposition 3 Let the operator L0 be the Hydrogen-atom operator with para-meters q0 � 3

4 and q1 ∈ R, that is, on the domain (19) it acts as

(L0 y)(x) = −y′′(x) +(q0

x2+ q1

x

)y(x).

Then the following holds

σc(L0) = [0, ∞)

and

σp(L0) =

⎧⎪⎪⎨⎪⎪⎩

∅ if q1 � 0{λ j := −q2

1(2 j − 1 + 2

√q0 + 1

4

)2

∣∣∣ j = 1, 2, . . .}

if q1 < 0.

Remark 10 The spectrum is simple.

Proof Here and in the following the function√· has its branch cut on the

negative half line. For λ = 0 substituting z := −2√−λ x in the eigenvalue

equation

(L0 y)(x) = −y′′(x) +(q0

x2+ q1

x

)y(x) = λy(x) (57)

yields the Whittaker equation (56) with parameters m =√

14 + q0 and l = q1

2√−λ

and hence (57) has the linearly independent solutions

y1(x, λ) := W q12√−λ

,√

14 +q0

(− 2√−λ x

)

and

y2(x, λ) := W− q12√−λ

,√

14 +q0

(2√−λ x

).

For λ > 0 the estimate (55) implies that neither y1 nor y2 vanish as x → ∞.It is easy to see that the same holds true for all their linear combinations.Hence

σp(L0) ∩ R+ = ∅.


From the asymptotics at ∞ it also follows that for Re λ > 0 the Weyl solutiong( · , λ) is a multiple of either y1( · , λ) or y2( · , λ) depending on the sign of Im λ.However, due to the fact that these functions have different asymptotics at ∞it is easy to see that g( · , λ) has a (finite) jump as λ crosses the real line andhence

(0, ∞) ⊂ σc(L0).

In the same way one sees that on the negative real line there are no jumps(with the possible exception of a discrete set of points) and hence there is nocontinuous spectrum in (−∞, 0). Let us now consider λ = 0. For q1 = 0 it iseasy to check that every solution of (57) is given by

y(x) := √x Hγ (2i

√q1

√x),

where the function Hγ (z) is an arbitrary solution of the Bessel equationd2

dz2 H(z) + 1z

ddz H(z) + (1 − γ 2

z2 )H(z) = 0 with γ := 2√

14 + q0. Using the as-

ymptotics of Bessel functions one can check that for q1 < 0 no solution issquare integrable at ∞, and for q1 > 0 there exists a solution which is squareintegrable at ∞, however, not at x = 0. Furthermore, one can see directly thatalso for q1 = 0 the solution of (57) which is square integrable at ∞ does notbelong to L2(0, ∞) due to its singularity at the origin. Hence in any case

0 ∈ σp(L0).

Note that for q1 � 0 the operator L0 is non-negative and hence it holds R− ⊂�(L0). If q1 < 0, however, there exists a sequence of negative eigenvalues,accumulating at λ = 0. See eg. [14, 18, 29] for how to determine this sequenceexplicitly. �

As a direct consequence of Proposition 3 we also describe the spectraof Lθ .

Theorem 3 Let the operator Lθ for θ ∈ [0, π) be given as in Section 3.3. Then

σc(Lθ ) = [0, ∞).

– If q1 � 0 then σp(Lθ ) consists of at most f initely many negative points. Thereexists exactly one exceptional parameter θ0 ∈ (0, π) such that 0 ∈ σp(Lθ0).

– If q1 < 0 then σp(Lθ ) consists of a sequence of negative points accumulatingat 0, but λ = 0 is never an eigenvalue.

Proof Since for each θ ∈ [0, π) the operator Lθ is a finite rank perturbation (inthe resolvent sense) of L0 it follows from Proposition 3 that

R+ ⊂ σc(Lθ ) ∪ σp(Lθ )


and

R− ⊂ �(Lθ ) ∪ σp(Lθ )

and, in particular, that for q1 � 0 the number of negative eigenvalues has to befinite.

Assume now that λ0 > 0 is an eigenvalue of some Lθ with eigenelementU0. Then according to Remark 8 the embedding �(U0) satisfies equation (57).However, in the proof of Proposition 3 we have seen that none of thesesolutions are square integrable at ∞. Hence for each parameter θ ∈ [0, π) thisimplies R

+ ⊂ σc(Lθ ).

Let us now consider the point λ = 0. As before for q1 < 0 there is no solutionof (57) which is square integrable at ∞. However, for q1 � 0 there is such asolution u0. Using Lemma 4 it is not hard to see that this function actuallybelongs to the range of �(H). Then obviously for exactly one value θ0 ∈ (0, π)

the boundary condition (36) is satisfied for U0 := �−1u0. Hence the point λ = 0is an eigenvalue exactly for the operator Lθ0 . �

The above considerations provide also more insight in the behavior of m(λ)

and g( · , λ) towards the positive real line.

Corollary 3 The functions m(λ) and − 1

Q(λ) + cot θ, which are analytic in the

upper half plane, can be continued analytically to every point λ0 ∈ R+.

Remark 11

(i) Note, in particular, that for every fixed x ∈ R also g(x, λ) has an analyticcontinuation across the positive real line.

(ii) In general, these continuations will not be real valued on R+.

(iii) On the lower half plane the continuation does in general not coincidewith the original function.

Proof Since the function y1(x, λ) in the proof of Proposition 3 is a solution of(57) which is square integrable at ∞ for each λ with Im λ > 0 it is proportionalto g(x, λ), i.e.

y1(x, λ) = a(λ)g(x, λ) = a(λ)[g−(x, λ) − m(λ)g+(x, λ)

]

holds with some factor a(λ). Note that y1(x, λ) and ddx y1(x, λ) are both holo-

morphic at least for λ ∈ C \ R−0 . Since g+(x, λ) and d

dx g+(x, λ) are entire in λ itfollows that a(λ), which can be expressed via the Wronskian by

a(λ) = W(y1( · , λ), g+( · λ)

),

can be continued holomorphically across R+. Hence, since obviously a(λ) = 0,

also g(x, λ) = y+(x,λ)

a(λ)can be continued holomorphically across the positive real


line. Note that for every λ0 ∈ R+ there exists an x0 ∈ R

+ such that g+(x0, λ0) =0 and hence also

m(λ) = g−(x0, λ) − g(x0, λ)

g+(x0, λ)

can be continued holomorphically to λ0. Therefore by Theorem 2 also Q(λ) +cot θ can be continued. However, note that this continuation cannot vanish inany λ0 ∈ R

+. Indeed, due to the analyticity then also the limit

limλ→λ0

Q(λ) + cot θ

λ − λ0

would exist, i.e. λ0 is a zero of the function Q(λ) + cot θ , which would implyλ0 ∈ σp(Lθ ). Hence, finally, also − 1

Q(λ)+cot θ has an analytic continuation acrossR

+. �

Alternatively to the Whittaker functions one can also use another set oflinearly independent solutions of (56), the so-called Kummer-functions, whichsatisfy as z → 0

Ml,m(z) = z12 +me− z

2

(1 +

12 + m − l

1!(2m + 1)z+(

12 + m − l

)(32 + m − l

)2!(2m + 1)(2m + 2)

z2 + . . .

),

Ml,−m(z) = z12 −me− z

2

(1 +

12 − m − l

1!(−2m + 1)z +

(12 − m − l

)(32 − m − l

)2!(−2m + 1)(−2m + 2)

z2 + . . .

).

If 2m ∈ N then the following relation holds

Wl,m(z) = �(−2m)

�(

12 − m − l

)Ml,m(z) + �(2m)

�(

12 + m − l

)Ml,−m(z), (58)

where � denotes the Gamma function. This relation between solutions withknown asymptotics at 0 and ∞, respectively, immediately implies the followingexplicit form of m.

Corollary 4 For 2√

14 + q0 = α+ − α− ∈ N the generalized Titchmarsh Weyl

coef f icient m has the form

m(λ)= 1

2√

14 + q0

·�

(12 +

√14 + q0 − q1

2√−λ

)

�

(12 −

√14 + q0 − q1

2√−λ

) ·�

(−2√

14 + q0

)

�

(2√

14 + q0

) ·(−4λ)√

14 +q0 ,

where the branch cut of the function z√

14 +q0 lies on the negative half line.


4.2 Standard and Generalized Spectral Representations

Finally we are going to use the spectral resolution of the identity for the modeloperator Lθ in order to show a new expansion result involving functions whichare not square integrable locally at 0.

In what follows we fix θ ∈ (0, π), note that θ = 0 corresponds to the standardcase, and we consider the model operator Lθ . Denote by N(θ) ∈ N ∪ {∞} thenumber of the negative eigenvalues of Lθ . In the case that 0 is an eigenvalue letthe function e0(x) denote ��0, the embedding of the eigenelement �0 for theeigenvalue 0, which actually can also be written in terms of Bessel functions.In the case 0 is not an eigenvalue let e0(x) ≡ 0. Moreover, on the positive halfline we define the function

�(x, λ) := g+(x, λ) − 1

d(λ + i0)g(x, λ + i0) λ ∈ R

+,

where the functions on the right hand side are to be understood as theanalytic continuation from the upper half plane, as in Corollary 3. We can nowformulate our main expansion theorem.

Theorem 4 With the above notations for every function U ∈ C∞0

((0, ∞)

)it

holds

U(x) =

⟨�0,

( 0U

)⟩H

〈�0, �0〉H

e0(x) +N(θ)∑j=1

g(x, λ j)

d′(λ j)

∫ ∞

0g(s, λ j)U(s) ds

+ 1

2π i

∫ ∞

0

∫ ∞

0�(x, λ)�(s, λ) U(s) ds (m(λ+i0) − m(λ−i0)) dλ. (59)

Remark 12 The functions �(x, λ) in this expansion are not locally squareintegrable on [0, ∞), except if θ = 0, where � = g+. In this special case theexpansions are well known in the literature (see eg. [19] or more recently[15, 17])

Remark 13 Expansion (59) is obviously dependent on the parameter θ , inparticular, the eigenvalues λ j and the ‘generalized eigenfunctions’ � (via thefunction d(λ + i0)).

As first step we are going to show the following auxiliary lemma, which is aweak version of the spectral decomposition corresponding to the operator Lθ

in the model space H.


In what follows the notation

〈F, G〉H = (f, �g)Cn−2 +∞∫

0

b n−2(L0)F(x)G(x) dx

is used in a natural way also for F =( f

F

)and G =

( gG

)for which the above

integral exists even if the functions F, G do not belong to Hn−2(L0). However,then it can be interpreted as a pairing with respect to the operator L0.

Lemma 5 Let elements U :=( u

U

)and V :=

( vV

)be given with U, V ∈

C∞0

((0, ∞)

). Then it holds

〈U, V〉H = 〈ELθ({0})U, V〉H +

N(θ)∑j=1

〈U, � j〉H〈� j, V〉H

〈� j, � j〉H

+ 1

2π i

∫ ∞

0〈U, �(λ)〉H〈�(λ), V〉H

m(λ+i0) − m(λ−i0)

b n−2(λ)dλ, (60)

where

1. the operator ELθ({0}) denotes the spectral projection corresponding to the

(possible) eigenvalue 0,2. the elements (� j)

N(θ)

j=1 are the eigenfunctions for the negative eigenvaluesof Lθ ,

3. the element � is def ined as � =( 0

g+

)+ 1

Q+cot θ �,

4. b n−2 is the polynomial introduced in (34).

Note that the integral in (60) should be understood as an improper integral.

Proof Recall that for the self-adjoint operator Lθ in the Hilbert space H Stonesformula (see [13, page 1203]) holds,

〈U, ELθ((λ1, λ2])V〉H = lim

δ↓0limε↓0

1

2π i

λ2+δ∫λ1+δ

(〈U, (Lθ − (λ + iε))−1V〉H

−〈U, (Lθ − (λ − iε))−1V〉H

)dλ,

(61)


where U, V are as in the formulation of the Lemma and ELθ((λ1, λ2]) denotes

the spectral projection for the operator Lθ corresponding to the interval(λ1, λ2]. In what follows we choose 0 < λ1 < λ2.

Using Krein’s formula (37) for the resolvent of Lθ and rearranging theintegrand we can write (61) as

limδ↓0

limε↓0

1

2π i

λ2+δ∫λ1+δ

[(u, �

((M − (λ + iε))−1 − (M − (λ − iε))−1)v)

Cn−2

+(

U, b n−2(L0)((L − (λ + iε))−1 − (L − (λ − iε))−1

)V)

L2(0,∞)

− 1

Q(λ + iε) + cot θ〈�(λ − iε) − �(λ + iε),V〉H〈U, �(λ + iε)〉H

+ 1

Q(λ − iε) + cot θ〈�(λ + iε),V〉H〈U, �(λ − iε) − �(λ + iε)〉H

+ Q(λ + iε) − Q(λ − iε)|Q(λ − iε) + cot θ |2 〈�(λ + iε),V〉H〈U, �(λ + iε)〉H

]dλ.

Since the first summand is holomorphic in the relevant interval the limit forε ↓ 0 of its contribution vanishes. In order to investigate the other terms notefirst that for any solution of the equation

−y′′ + q0 + q1xx2

y = λy,

which is holomorphic in λ, it holds

y(x, λ ± iε) = y(x, λ) + O(ε), (62)

where the term O(ε) is uniform with respect to (x, λ) on each compact subsetof (0, ∞) × R. Then the second summand

limε↓0

1

2π i

λ2+δ∫λ1+δ

(U, b n−2(L0)

((L − (λ + iε))−1 − (L − (λ − iε))−1

)V)

L2(0,∞)dλ

can be simplified by inserting the explicit form of the resolvents as inte-gral operators as given in Lemma 3, observing that the function m(λ) inthe relation g(x, λ) = g−(x, λ) − m(λ)g+(x, λ) has a holomorphic continuation


to the positive real line and, finally, using the uniform estimate (62). Thisyields

1

2π i

λ2+δ∫λ1+δ

∞∫0

(b n−2(L0)U(x))g+(x, λ) dx

·∞∫

0

g+(s, λ)V(s) ds(m(λ + i0) − m(λ − i0)

)dλ

= 1

2π i

λ2+δ∫λ1+δ

∞∫0

g+(x, λ)b n−2(L0)U(x) dx

·∞∫

0

g+(s, λ)b n−2(L0)V(s) dsm(λ + i0) − m(λ − i0)

b n−2(λ)dλ,

where the last equality uses that g+(x, λ) is a solution of (6). For the otherterms note that from the definition of � in (38) it follows for λ ∈ R

+

〈U, �(λ + iε)〉 = 〈U, �(λ + i0)〉 + O(ε)

and

〈�(λ − iε) − �(λ + iε), V〉

=∞∫

0

g+(s, λ)b n−2(L0)V(s) dsm(λ + i0) − m(λ − i0)

b n−2(λ)+ O(ε)

and for the other expressions correspondingly. Collecting all terms gives

〈U, ELθ((λ1, λ2])V〉H

= limδ↓0

1

2π i

λ2+δ∫λ1+δ

〈U, �(λ + i0)〉〈�(λ + i0), V〉m(λ + i0) − m(λ − i0)

b n−2(λ)dλ

= 1

2π i

λ2∫λ1

〈U, �(λ + i0)〉〈�(λ + i0), V〉m(λ + i0) − m(λ − i0)

b n−2(λ)dλ,

since σp(Lθ ) ∩ R+ = ∅. This implies further

〈U, ELθ((0, ∞))V〉H

= 1

2π i

∞∫0

〈U, �(λ + i0)〉〈�(λ + i0), V〉m(λ + i0) − m(λ − i0)

b n−2(λ)dλ


Finally the decomposition of the identity operator I as

I = ELθ({0}) +

N(θ)∑j=1

ELθ({λ j}) + ELθ

((0, ∞)),

where ELθ({λ j}) = 〈� j, · 〉

〈� j,� j〉� j is the orthogonal projection onto the eigenspacespanned by the eigenelement � j := �(λ j) corresponding to the eigenvalue λ j,yields the desired weak expansion. �

Using this result we are now going to prove Theorem 4.

Proof First we show that for every U =( 0

U

)with U ∈ C∞

0

((0, ∞)

)the

transformation

IU := ELθ({0})U +

N(θ)∑j=1

〈� j, U〉H

〈� j, � j〉H

� j

+ 1

2π i

∫ ∞

0〈�(λ), U〉H�(λ)

m(λ + i0) − m(λ − i0)

b n−2(λ)dλ (63)

is well defined, which means that we have to verify the existence of the integral.Let us start with the right endpoint +∞. Since U belongs to the domain of anypower N � 0 of the operator Lθ Lemma 5 can be applied to the inner product〈LN

θ U, U〉H. Integration by parts and using the eigenvalue property leads to

〈LNθ U, U〉H =

N(θ)∑j=1

λNj|〈� j, U〉H|2〈� j, � j〉H

+ 1

2π i

∫ ∞

0λN|〈�(λ), U〉H|2(Q(λ+i0)

−Q(λ−i0)) dλ, (64)

where we also used formula (42) and Theorem 2. Since Q as a Nevanlinnafunction grows at most linearly it follows that∫ ∞

0λN|〈�(λ), U〉H|2 dλ < ∞ for all N ∈ N,

and hence 〈�(λ), U〉H tends to zero faster than any power of λ.Let us now consider the second factor in the integral in (63)

�(λ) = (0)g+( · , λ) − 1

Q(λ + i0) + cot θ�(λ + i0).

The identity

g+(x, λ) = −g(x, λ + i0) − g(x, λ − i0)

m(λ + i0) − m(λ − i0)

together with Remark 11 and Corollary 4 imply that g+(x, λ) is (locally uni-formly in x) bounded in λ as λ → +∞. Similarly also ��(x, λ) = 1

b n−2(λ)g(x, λ)


is (locally uniformly in x) bounded for large real λ. Since the functions gk,which are used for the embedding, are independent of λ and bounded at ∞it follows that both components of �(λ) are (locally uniformly in x) boundedand hence the integral in (64) converges (locally uniformly in x) at +∞.

Let us next consider the left endpoint λ = 0. For U = V =( 0

U

)Lemma 5

implies that the integral∫ ∞

0|〈�(λ), U〉H|2(Q(λ+i0) − Q(λ−i0)) dλ

exists. Hence

〈�(λ), U〉H

√Im Q(λ+i0) ∈ L2(0, ∞).

So it remains to show that

�(λ)√

Im Q(λ+i0) ∈ L2(0, λ0) for some λ0 > 0, (65)

or, equivalently, that both the functions√

Im Q(λ+i0) and(

g+(x, λ) − 1

d(λ + i0)g(x, λ)

)√Im Q(λ+i0)

are integrable on the interval (0, λ0). Here the second function is just �� andthe other corresponds to the first component of �. Since g+ and g− are bothentire functions in λ it is sufficient for (65) that both the functions

√Im Q(λ+i0) and

√Im Q(λ+i0)

|Q(λ + i0) + cot θ | =√

Im−1

Q(λ + i0) + cot θ

belong to L2(0, λ0). However, this follows directly from the fact that thesefunctions are actually the densities of the spectral measures of the Nevanlinnafunctions Q and −1

Q , respectively. Note that here both functions have no gene-ralized poles in (0, ∞) and hence on this interval the measures are absolutelycontinuous with respect to the Lebesgue measure. Thus the transformation Iin (63) is well defined. Next we show that actually IU = U for all U =

( 0U

)

with U ∈ C∞0

((0, ∞)

). To this end note that Lemma 5 implies

〈IU, V〉H = 〈U, V〉H (66)

for all V =( v

V

)with V ∈ C∞

0

((0, ∞)

). Considering, particularly, V =

(v0

)implies that the first component of IU has to be the zero vector in C

n−2.Writing out (66) yields∫ ∞

0

((IU)Hn−2 − U

)(x) b n−2(L)V(x) dx = 0,

where the index Hn−2 denotes the second component in the space H. Sinceb n−2(L)V(x) can be an arbitrary C∞

0

((0,∞)

)function it follows that (IU)Hn−2


and U coincide as elements in L2. But then the local uniform convergenceof the integral in (63) implies that (IU)Hn−2(x) is a continuous function andhence

(IU)Hn−2(x) = U(x) for all x ∈ (0, ∞).

In what follows we apply the embedding � to the identity

U = ELθ({0})U +

N(θ)∑j=1

〈� j, U〉H

〈� j, � j〉H

� j

+ 1

2π i

∫ ∞

0〈�(λ), U〉H�(λ)

m(λ + i0) − m(λ + i0)

b n−2(λ)dλ.

Then the left hand side obviously becomes �U = U . On the right hand sidenote that due to (41) for every eigenvalue λ j < 0 it holds

�� j = ��(λ j) = 1

b n−2(λ j)g( · , λ j)

and with the natural extension of � to the generalized eigenfunctions oneobtains

��(λ) = �(λ).

Note, furthermore,

〈� j, U〉H =∫ ∞

0(L − λ j)

−1gn−2(x)(b n−2(L)U)(x)dx =∫ ∞

0g(x, λ j)U(x)dx

and

〈�(λ), U〉H =∫ ∞

0b n−2(λ)�(x, λ)U(x)dx =

∫ ∞

0g(x, λ j)U(x)dx.

Using the identity

b n−2(λ j)〈� j, � j〉H = b n−2(λ j)Q′(λ j) = d′(λ j)

finally implies expansion (59). �

5 Concluding Remarks

Our approach using supersingular perturbations heavily relies on the asymp-totic behavior of the special solutions g+ and g− close to the origin. However,besides the fact that (6) is in limit point case at ∞, these are the only propertiesof the differential expression (1) that are really used in the construction of themodel (Sections 2 and 3). Hence the results actually hold true also for moregeneral holomorphic potentials.


The natural question, which arises next, is whether the construction can alsowork for the even wider class of potentials considered in [17]. This, however,is work in progress.

Acknowledgements We thank the referee for careful reading of the manuscript and usefulcritical comments.

Appendix A: The Scale of Hilbert Spaces

Recall that if A is a semi-bounded, self-adjoint linear operator in a Hilbertspace H, A � γ for some γ ∈ R, then the scale of spaces Hs(A) associatedwith A is defined as follows. For s � 0 the space Hs(A) is given by the setdom (A − μ)

s2 equipped with the norm

‖y‖s := ‖(A − μ)s2 y‖H (67)

for some μ < γ . However, it can easily be seen that this definition does notdepend on μ and furthermore Hs(A) is complete with this norm. The spaceH−s(A) is then defined as the dual of Hs(A) (with respect to the original spaceH), it can also be obtained by completing H with respect to the norm (67).This gives a scale of spaces Hs(A) ⊂ Ht(A) if s > t. However, in this note weare dealing with s ∈ Z only:

...⊂H3(A) ⊂dom (A)

‖H2(A) ⊂H1(A) ⊂

H‖H0(A) ⊂H−1(A) ⊂

(dom (A))∗‖H−2(A) ⊂ H−3(A) ⊂...

The notation 〈 · , · 〉 is used not only for the usual inner product on the space H,but 〈g, f 〉 denotes also the action of the functional f ∈ H−s(A) on an elementg ∈ Hs(A), s > 0. Note that (A − μ)− t

2 can be seen as an isometry from Hs(A)

to Hs+t. In particular, 〈g, (A − μ)t2 f 〉 for g ∈ Hs+t and f ∈ H−s is given by

〈(A − μ)t2 g, f 〉.

Appendix B: Limit Circle Case

For completeness reasons and in order to establish a connection to the forego-ing considerations in the ‘singular’ situation, we want to recall briefly also the‘regular’ case − 1

4 � q0 < 34 , in which Lemma 1 implies that the expression � is

in limit circle case also at the endpoint 0, see [15, 17] and e.g. [2, 27] for generalresults on rank one perturbations.

With the differential expression (1)

�(y)(x) = −y′′(x) + q0 + q1xx2

y(x) on x ∈ (0, ∞)

there is associated the maximal operator Lmax by

dom Lmax := {y∈ L2(0, ∞)|y, y′ ∈ ACloc(0, ∞), �(y)∈ L2(0, ∞)} (68)


and

(Lmax y)(x) := �(y)(x).

Since here we assume that � is in limit circle case at one and in limit pointcase at the other endpoint the maximal operator Lmax is the adjoint of asymmetric minimal operator Lmin with defect one. Its domain is given by

dom Lmin ={

y ∈ dom Lmax | limx→0

W(y(x), g+(x, λ0))

= 0 limx→0

W(y(x), g−(x, λ0)) = 0}.

In this case g( · , λ) ∈ L2(0, ∞) is a defect element of Lmin. Note that, in fact,Lmin does not depend on the particular choice of Iλ0.

Remark 14 In case x = 0 is a regular endpoint the two expressionslimx→0

W(y(x), g+(x, λ0)) and limx→0

W(y(x), g−(x, λ0)) can be written as y(0) and

y′(0), respectively.

The self-adjoint extensions of Lmin are given by Lτ for τ ∈ R with

dom Lτ :={

y ∈ dom Lmax| limx→0

W(y(x), g+(x, λ0) + τg−(x, λ0)) = 0}

and

dom L∞ :={

y ∈ dom Lmax| limx→0

W(y(x), g−(x, λ0)) = 0}.

It is then a standard calculation to show that the following integral formulasfor the resolvents hold:((Lτ − λ)−1 y

)(x)=−g(x, λ)

∫ x

0gτ (s, λ)y(s) ds − gτ (x, λ)

∫ ∞

xg(s, λ)y(s) ds,

where gτ ( · , λ) := 11−τm(λ)

(g+( · , λ) − τg−( · , λ)) for τ ∈ R and in the limit caseg∞( · , λ) := 1

m(λ)g−( · , λ). Furthermore, from this one obtains directly

(Lτ − λ)−1 = (L0 − λ)−1 − 1

m(λ) − 1τ

〈g(x, λ), · 〉g(x, λ),

where 〈 · , · 〉 denotes the usual inner product in L2(0, ∞).We introduce now the singular element ϕ := (L0 − λ0)g( · , λ0) for some

λ0 ∈ C. Note that ϕ ∈ L2(0, ∞) since g ∈ dom L0, however, ϕ ∈ H−2(L0).With the formal expression

L0 + t〈ϕ, · 〉ϕ t ∈ R ∪ {∞}there is associated a family of self-adjoint operators Lγ given by

(Lγ − λ)−1 = (L0 − λ)−1 − 〈(L0 − λ)−1ϕ, · 〉Q(λ) + γ

(L0 − λ)−1ϕ, γ ∈ R ∪ {∞},


where the Q-function is defined as

Q(λ) := (λ − μ1)〈ϕ, (L0 − λ)−1(L0 − μ1)−1ϕ〉

with some μ1 ∈ ρ(L0) ∩ R. It can easily be checked that Q(λ) = m(λ) + c withsome constant c ∈ R, which fits as a well known special case to Theorem 2.

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The Nonlinear Schrödinger Equationwith a Self-consistent Source in the Classof Periodic Functions

Alisher Yakhshimuratov

Received: 4 August 2010 / Accepted: 28 February 2011 / Published online: 31 March 2011© Springer Science+Business Media B.V. 2011

Abstract In this work the method of inverse spectral problem is applied to theintegration of the nonlinear Schrödinger equation with a self-consistent sourcein the class of periodic functions.

Keywords Dirac’s operator · Spectral data · The system of equations ofDubrovin–Trubowitz · Nonlinear Schrödinger equation with a self-consistentsource

Mathematics Subject Classifications 2010 39A70 · 37K15 · 37K60 · 35Q53

1 Introduction

One of the representative of the class of completely integrable nonlinear par-tial differential equations, which has a great practical importance, is the non-linear Schrödinger (NLS) equation. Complete integrability of this equation,for the class of rapidly decreasing functions, has been established, by meansof the inverse problem method, for the first time in [1–4]. In the monographs[5–8], which are devoted to the integration of nonlinear equations, for the NLSequation were given special attention. For the investigation of NLS equation inthe class of the periodic or finite-gap functions the reader is referred to [9–22].

This paper is devoted to the studying of the nonlinear Schrödinger equationwith a self-consistent source in the class of periodic functions. We have tonote that the NLS equation with a self-consistent source, in the class of

A. Yakhshimuratov (B)Department of Mathematics, Urgench State University, 14 Kh. Alimdjan, 220100, Urgench,Uzbekistane-mail: [email protected]

154 A. Yakhshimuratov

rapidly decreasing functions, was considered in [23–26]. Moreover, nonlinearequations with a self-consistent source in the class of the periodic functionswere studied for different kind of problems in [27, 28].

2 Statement of the Problem

We study the following NLS equation with self-consistent source

ut = 2iu |u|2 − iuxx + b(t)ux

+∞∑

k=−∞iαk(t)s1(π, λk, t) (ψ1(x, λk, t) − iψ2(x, λk, t))2

+∞∫

−∞iβ(λ, t)s1(π, λ, t)(ψ1 − iψ2)(ψ1 − iψ2)dλ, t > 0, x ∈ R1 (1)

coupled with the initial condition

u(x, t)|t=0 = u0(x). (2)

We look for complex-valued solution that is π -periodic on the partial variablex and satisfy the following regularity assumptions:

u(x, t) ∈ C2x(t > 0) ∩ C1

t (t > 0) ∩ C(t ≥ 0). (3)

Here αk(t), k ∈ Z is a given sequence of continuous real functions hav-ing a uniform asymptotic behavior αk(t) = O( 1

k2 ), k → ±∞, b(t) is a givenreal continuous function, β(λ, t) is a given real continuous function hav-ing a uniform asymptotic behavior β(λ, t) = O( 1

λ2 ), λ → ±∞ and ψ =(ψ1(x, λ, t), ψ2(x, λ, t))T is the Floquet’s solution (normalized by conditionψ1(0, λ, t) = 1) of the Dirac’s equation

L(t)y ≡ Bdydx

+ �(x, t)y = λy, x ∈ R1. (4)

Here

B =(

0 1−1 0

), �(x, t) =

(p(x, t) q(x, t)q(x, t) −p(x, t)

), y =

(y1(x)

y2(x)

),

and p(x, t) = −Re(u(x, t)), q(x, t) = Im(u(x, t)).We denote by λk, k ∈ Z the sequence of eigenvalues of the Eq. 4 with

the periodic (y(π) = y(0)) or antiperiodic (y(π) = −y(0)) boundary valueconditions (λ4k−1, λ4k, k ∈ Z correspond to a periodic boundary condition,and λ4k+1, λ4k+2, k ∈ Z correspond to antiperiodic boundary condition). Thenumbering introduced in such a way, where ξn(t) ∈ [λ2n−1, λ2n], n ∈ Z and

The Nonlinear Schrödinger Equation with a Self-consistent Source... 155

ξn(t) = n + o(1), (n → ∞). Here ξn(t), n ∈ Z are eigenvalues of the Eq. 4coupled with the Dirichlet boundary conditions

y1(0) = 0, y1(π) = 0. (5)

Note, these eigenvalues coincide with the zeros of the function s1(π , λ, t),where s1(x, λ, t) is the first component of the solution s(x, λ, t) of the Eq. 4satisfying the initial condition s(0, λ, t) = (0, 1)T .

The aim of this work is to develop a procedure for constructing the solu-tion (u(x, t), ψ(x, λ, t)) of problem (1)–(4), by means of the inverse spectralproblem, related to the Dirac’s Eq. 4.

3 Preliminaries

In this section we present some of the basic properties of Dirac operator forthe sake of completeness. The reader can find detailed information on spectralproperties of the Dirac operator in [29–36] and references therein.

Consider the following system of Dirac equations on the whole line

Ly ≡(

0 1−1 0

)(y′

1y′

2

)+(

p(x) q(x)

q(x) −p(x)

)(y1

y2

)= λ

(y1

y2

), x ∈ R1, (6)

where p(x) and q(x) are two real continuous and π -periodic functions, and λ

is a complex parameter.We denote by c(x, λ) and s(x, λ) the unique solutions of (6), which satisfy

the initial conditions c(0, λ) = (1, 0)T and s(0, λ) = (0, 1)T , respectively.The function (λ) = c1(π, λ) + s2(π, λ) is called as Lyapunov’s function or

Hill’s discriminant of system (6).The spectrum of problem (6) has the form

σ(L) = {λ ∈ R1

∣∣ − 2 � (λ) � 2} = R1\

∞⋃

n=−∞(λ2n−1, λ2n).

The intervals (λ2n−1, λ2n), n ∈ Z are called as gaps or lacunas.Let ξn, n ∈ Z be the roots of the equation s1(π, λ) = 0. We note that ξn, n ∈

Z , coincide with the eigenvalues of the Dirichlet problem y1(0) = 0, y1(π) = 0for system (6), moreover, the following inclusions ξn ∈ [λ2n−1, λ2n], n ∈ Z arefulfilled.

The numbers ξn ∈ [λ2n−1, λ2n], n ∈ Z , with signs σn = sign{1 − s22(π, ξn)}, n ∈

Z , are called as the spectral parameters of the problem (6).The spectral parameters ξn, σn, n ∈ Z and the boundaries of the spectrum

we call as the spectral data of problem (6).The finding of the spectral data of the problem (6) is called as the direct

problem and conversely, restoration of the coefficients p(x) and q(x) of theproblem (6), by means of the spectral data, is called as the inverse problem.


The spectrum of Dirac’s operator, with coefficients p(x + τ) and q(x + τ),does not depend on the real parameter τ , but spectral parameters depend fromτ : ξn(τ ), σn(τ ), n ∈ Z . These spectral parameters satisfy the following systemof differential equations of Dubrovin–Trubowitz

dξn

dτ= σn(τ )

√(ξn − λ2n−1)(λ2n − ξn) ·

√√√√√√

∞∏

k = −∞,

k = n

(λ2k−1 − ξn)(λ2k − ξn)

(ξk − ξn)2

×

⎧⎪⎪⎪⎨

⎪⎪⎪⎩λ2n−1 + λ2n +

∞∑

k = −∞,

k = n

(λ2k−1 + λ2k − 2ξk)

⎫⎪⎪⎪⎬

⎪⎪⎪⎭, n ∈ Z .

The system of equations Dubrovin–Trubowitz and the following trace formu-las (see [37], p. 97)

p(τ ) =∞∑

k=−∞

(λ2k−1 + λ2k

2− ξk(τ )

),

q2(τ ) + q′(τ ) =∞∑

k=−∞

(λ2

2k−1 + λ22k

2− ξ 2

k (τ )

),

q(τ ) =∞∑

n=−∞σn(τ )

√(ξn(τ ) − λ2n−1) (λ2n − ξn(τ ))

·√√√√√√

∞∏

k = −∞k = n

(λ2k−1 − ξn(τ )) (λ2k − ξn(τ ))

(ξk(τ ) − ξn(τ ))2 ,

provide a method of solving of the inverse problem.We note that the previous formula can be proved by applying the Mittag–

Leffler’s theorem to the function s2(π,λ)−c1(π,λ)

2s1(π,λ).

4 Main Result

Now, we are in a position to prove the main result of this work. The result isthe following


Theorem Let (u(x, t), ψ(x, λ, t)) be the solution of the problem (1)–(4). Thenthe spectrum of Eq. 4 does not depend on t, and the spectral parameters ξn(t),n ∈ Z satisfy the system of Dubrovin–Trubowitz’s equations:

ξn(t) = −2σn(t)√

(ξn − λ2n−1)(λ2n − ξn) ·√√√√√√

∞∏

k = −∞,

k = n

(λ2k−1 − ξn)(λ2k − ξn)

(ξk − ξn)2

×

⎧⎪⎨

⎪⎩q2(0, t) + qx(0, t) + [

p(0, t) + ξn]2 + ξ 2

n − b(t)[

p(0, t) + ξn]

+∞∑

k=−∞

αk(t)s1(π, λk, t)ξn − λk

+∞∫

−∞

β(λ, t)s1(π, λ, t)ξn − λ

dλ

⎫⎪⎬

⎪⎭. (7)

The signs σn(t) = ±1 change in each collision of the function ξn(t) with theboundaries of its gap [λ2n−1, λ2n]. Moreover, the following initial conditions arefulf illed:

ξn(t)|t=0 = ξ 0n , σn(t)|t=0 = σ 0

n , n ∈ Z ,

where ξ 0n , σ 0

n , n ∈ Z are the spectral parameters of the Dirac’s equation corre-sponding to the coef f icients p0(x) = −Re(u0(x)) and q0(x) = Im(u0(x)).

Proof Using the equality u(x, t) = −p(x, t) + iq(x, t) and introducing the fol-lowing notations

G1(x, t) = −2∞∑

k=−∞αk(t)s1(π, λk, t)ψ1(x, λk, t)ψ2(x, λk, t)

−∞∫

−∞β(λ, t)s1(π, λ, t)(ψ1ψ2 + ψ2ψ1)dλ,

and

G2(x, t) =∞∑

k=−∞αk(t)s1(π, λk, t)

[ψ2

1 (x, λk, t) − ψ22 (x, λk, t)

]

+∞∫

−∞β(λ, t)s1(π, λ, t)(ψ1ψ1 − ψ2ψ2)dλ,


we rewrite Eq. 1 in the following form{

pt = 2q · (p2 + q2) − qxx + b px + G1

qt = −2p · (p2 + q2) + pxx + bqx + G2. (8)

Let yn(x, t) =(

yn,1(x, t)yn,2(x, t)

), n ∈ Z be the orthonormal eigenfunctions of the

Dirichlet problem (4)–(5), that corresponds to the eigenvalues ξn(t), n ∈ Z(see [29, ch. 7] for details). Without lost of generality, we may assume thatthe eigenfunctions yn(x, t), n ∈ Z , are real-valued.

Differentiating the identity ξn(t) = (L(t)yn, yn) with respect to t, and usingthe symmetry of the operator L(t), we arrive at the following expressions

ξn(t) = (�(x, t)yn + L(t)yn, yn

) + (L(t)yn, yn)

= (�(x, t)yn, yn

) + (yn, L(t)yn) + (L(t)yn, yn)

= (�(x, t)yn, yn

) + ξn(t)((yn, yn)) = (�(x, t)yn, yn

). (9)

On the other hand, by means of the explicit form of the scalar product (y, z) =π∫

0

[y1(x)z1(x) + y2(x)z2(x)

]dx, being y =

(y1(x)

y2(x)

)and z =

(z1(x)

z2(x)

), equality

(9) can be rewritten as

ξn(t) =π∫

0

[(pt yn,1 + qt yn,2)yn,1 + (qt yn,1 − pt yn,2)yn,2

]dx

=π∫

0

[(y2

n,1 − y2n,2

)pt + 2yn,1 yn,2qt

]dx. (10)

Substituting (8) into equality (10), we attain the following equality

ξn(t) =π∫

0

2(

p2 + q2) [(

y2n,1 − y2

n,2

)q − 2yn,1 yn,2 p

]dx

+π∫

0

[− (y2

n,1 − y2n,2

)qxx + 2yn,1 yn,2 · pxx

]dx

+π∫

0

b(t)[(

y2n,1 − y2

n,2

)px + 2yn,1 yn,2 · qx

]dx

+π∫

0

[(y2

n,1 − y2n,2

)G1 + 2yn,1 yn,2 · G2

]dx. (11)


Rewriting the first integral in equality (11) as

I1 =π∫

0

2(

p2 + q2) [

yn,1(q yn,1 − p yn,2) − yn,2(p yn,1 + q yn,2)]

dx (12)

and taking into account that Eq. 4 can be written as the following system

{y′

n,1 + ξn yn,2 = q yn,1 − p yn,2

ξn yn,1 − y′n,2 = p yn,1 + q yn,2

, (13)

we deduce the equality

I1 =π∫

0

2(

p2 + q2) [

yn,1(y′

n,1 + ξn yn,2) − yn,2

(ξn yn,1 − y′

n,2

)]dx

=π∫

0

(p2 + q2

) [y2

n,1 + y2n,2

]′dx =

π∫

0

(p2 + q2

)d(y2

n,1 + y2n,2

)

= (p2 + q2

) (y2

n,1 + y2n,2

)∣∣π0

−π∫

0

(y2

n,1 + y2n,2

)d(

p2 + q2)

= [p2(0, t) + q2(0, t)

] [y2

n,2(π, t) − y2n,2(0, t)

] −π∫

0

(y2

n,1 + y2n,2

)d(

p2 + q2).

(14)

Now, denote by I2 the second integral in equality (11). We have that

I2 = −π∫

0

(y2

n,1 − y2n,2

)dqx +

π∫

0

2yn,1 yn,2dpx

= (y2

n,2 − y2n,1

)qx|π0 +

π∫

0

2(yn,1 y′

n,1 − yn,2 y′n,2

)qxdx

+ 2yn,1 yn,2 px∣∣π0 −

π∫

0

2(y′

n,1 yn,2 + yn,1 y′n,2

)pxdx.


Using the system (13) again, we conclude that

I2 = qx(0, t)[y2

n,2(π, t) − y2n,2(0, t)

]

+π∫

0

2[yn,1(qyn,1 − pyn,2 − ξn yn,2) − yn,2(ξn yn,1 − pyn,1 − qyn,2)

]qxdx

−π∫

0

2[yn,2(qyn,1 − pyn,2 − ξn yn,2) + yn,1(ξn yn,1 − pyn,1 − qyn,2)

]pxdx

= qx(0, t)[y2

n,2(π, t) − y2n,2(0, t)

] +π∫

0

(y2

n,1 + y2n,2

)(p2 + q2)

−π∫

0

4ξn yn,1 yn,2dq −π∫

0

2ξn(y2

n,1 − y2n,2

)dp.

By integration by parts in the two last integrals, we get

I2 = qx(0, t)[y2

n,2(π, t) − y2n,2(0, t)

] +π∫

0

(y2

n,1 + y2n,2

)d(p2 + q2)

+ 2ξn p(0, t)[y2

n,2(π, t) − y2n,2(0, t)

]

+π∫

0

4ξn[y′

n,1(qyn,2 + pyn,1) + y′n,2(qyn,1 − pyn,2)

]dx.

Hence, using equalities (13) once again, we conclude that

I2 = qx(0, t)[y2

n,2(π, t) − y2n,2(0, t)

] +π∫

0

(y2

n,1 + y2n,2

)d(

p2 + q2)

+ 2ξn p(0, t)[y2

n,2(π, t) − y2n,2(0, t)

]

+π∫

0

4ξn[y′

n,1

(ξn yn,1 − y′

n,2

) + y′n,2

(y′

n,1 + ξn yn,2)]

dx

= qx(0, t)[y2

n,2(π, t) − y2n,2(0, t)

] +π∫

0

(y2

n,1 + y2n,2

)d(p2 + q2)

+ 2ξn p(0, t)[y2

n,2(π, t) − y2n,2(0, t)

] +π∫

0

2ξ 2n

(y2

n,1 + y2n,2

)′dx,


that is,

I2 = qx(0, t)[y2

n,2(π, t) − y2n,2(0, t)

] +π∫

0

(y2

n,1 + y2n,2

)d(

p2 + q2)

+ 2ξn p(0, t)[y2

n,2(π, t) − y2n,2(0, t)

] + 2ξ 2n

[y2

n,2(π, t) − y2n,2(0, t)

].

(15)

Integrating by parts and using (13) again, we have that

I3 =π∫

0

b(t)[(

y2n,1 − y2

n,2

)px + 2yn,1 yn,2 · qx

]dx

= −b(t)[

p(0, t) + ξn] [

y2n,2(π, t) − y2

n,2(0, t)]. (16)

Now we need to calculate the fourth integral in (11):

I4 =π∫

0

[(y2

n,1 − y2n,2

)G1 + 2yn,1 yn,2 · G2

]dx

=∞∑

k=−∞αk(t)s1(π, λk, t)

π∫

0

{−2(y2

n,1 − y2n,2

)ψ1(x, λk, t)ψ2(x, λk, t)

+ 2yn,1 yn,2[ψ2

1 (x, λk, t) − ψ22 (x, λk, t)

]}dx

+∞∫

−∞β(λ, t)s1(π, λ, t)

⎧⎪⎨

⎪⎩

π∫

0

[− (y2

n,1 − y2n,2

)(ψ1ψ2 + ψ2ψ1)

+ 2yn,1 yn,2(ψ1ψ1 − ψ2ψ2)]

dx

⎫⎪⎬

⎪⎭dλ. (17)

It is not difficult to verify that

J1 =π∫

0

{−2(y2

n,1 − y2n,2

)ψ1(x, λk, t)ψ2(x, λk, t)

+ 2yn,1 yn,2[ψ2

1 (x, λk, t) − ψ22 (x, λk, t)

]}dx

= −2

π∫

0

[y1,n · ψ1(x, λk, t) + y2,n · ψ2(x, λk, t)

] ·

· [y1,n · ψ2(x, λk, t) − y2,n · ψ1(x, λk, t)]

dx,


and

J2 =π∫

0

[− (y2

n,1 − y2n,2

)(ψ1ψ2 + ψ2ψ1) + 2yn,1 yn,2(ψ1ψ1 − ψ2ψ2)

]dx

= −π∫

0

[(y1,n · ψ1 + y2,n · ψ2) · (y1,n · ψ2 − y2,n · ψ1)

+ (y1,n · ψ1 + y2,n · ψ2) · (y1,n · ψ2 − y2,n · ψ1)]

dx.

Using the identity

(y1,n · ψ2 − y2,n · ψ1)′ = (λ − ξn)(y1,n · ψ1 + y2,n · ψ2)

we obtain

J1 = 1

ξn − λk· [y2

n,2(π, t) − y2n,2(0, t)

], (18)

J2 = 1

ξn − λ· [y2

n,2(π, t) − y2n,2(0, t)

]. (19)

Substituting (18) and (19) into (17) we conclude that

I4 =⎧⎨

⎩

∞∑

k=−∞


+∞∫

−∞


dλ

⎫⎬

⎭ · [y2n,2(π, t) − y2

n,2(0, t)].

(20)So, by means of expressions (11), (14), (15), (16) and (20) we deduce that

ξn(t) = [y2

n,2(π, t) − y2n,2(0, t)

]

×

⎧⎪⎨

⎪⎩q2(0, t) + qx(0, t) + [

p(0, t) + ξn]2 + ξ 2

n − b(t)[

p(0, t) + ξn]

+∞∑

k=−∞


+∞∫

−∞


dλ

⎫⎪⎬

⎪⎭. (21)

Using the equalities

s21(x, λ, t) + s2

2(x, λ, t) = sT · s = sT ·(

s + λ∂s∂λ

)− ∂sT

∂λ· (λs)

= sT ·(

B · ∂s′

∂λ+ � · ∂s

∂λ

)− ∂sT

∂λ· (B · s′ + � · s)

= sT · B · ∂s′

∂λ− ∂sT

∂λ· B · s′ =

(s1 · ∂s2

∂λ− s2 · ∂s1

∂λ

)′


we deduce the identityπ∫

0

[s2

1(x, λ, t) + s22(x, λ, t)

]dx = s1(π, λ, t) · ∂s2(π, λ, t)

∂λ

− s2(π, λ, t) · ∂s1(π, λ, t)∂λ

.

Hence, we find the norm of the eigenfunction s(x, ξn(t), t), corresponding tothe eigenvalue ξn(t) of the Dirichlet problem (4)–(5):

c2n(t)=

π∫

0

[s2

1 (x, ξn(t), t) + s22 (x, ξn(t), t)

]dx =− s2 (π, ξn(t), t) · ∂s1 (π, ξn(t), t)

∂λ.

(22)In particular, it follows that

sign{

∂s1 (π, ξn(t), t)∂λ

}= −sign {s2 (π, ξn(t), t)} . (23)

Using the equality

yn(x, t) = 1

cn(t)s (x, ξn(t), t)

and (22), we obtain

y2n,2(π, t) − y2

n,2(0, t) = s22 (π, ξn(t), t) − 1

c2n(t)

=1

s2(π,ξn(t),t) − s2 (π, ξn(t), t)∂s1(π,ξn(t),t)

∂λ

. (24)

Now, substituting the values x = π and λ = ξn(t) into identity

c1(x, λ, t)s2(x, λ, t) − c2(x, λ, t)s1(x, λ, t) = 1,

we can find that

c1 (π, ξn(t), t) = 1

s2 (π, ξn(t), t). (25)

With the help of (25) and the following identity

[c1(π, λ, t) − s2(π, λ, t)]2 = (2(λ) − 4

) − 4c2(π, λ, t)s1(π, λ, t),

we obtain the equality

1

s2 (π, ξn(t), t)− s2 (π, ξn(t), t) = sign

{1 − s2

2 (π, ξn(t), t)}

sign {s2 (π, ξn(t), t)}√

2 (ξn(t)) − 4,

(26)where (λ) = c1(π, λ, t) + s2(π, λ, t).

From (23), (24) and (26) we deduce equality

y2n,2(π, t) − y2

n,2(0, t) = −σn(t) ·√√√√

2 (ξn(t)) − 4(

∂s1(π,ξn(t),t)∂λ

)2 , (27)


where σn(t) = sign{1 − s2

2 (π, ξn(t), t)}.

Using expansions

2(λ) − 4 = −4π2∞∏

k=−∞

(λ − λ2k−1)(λ − λ2k)

a2k

, s1(π, λ, t) = π

∞∏

k=−∞

ξk − λ

ak,

where a0 = 1 and ak = k if k = 0, the identity (27) can be rewritten as:

y2n,2(π, t) − y2

n,2(0, t) = −2σn(t)√

(ξn − λ2n−1)(λ2n − ξn)

·√√√√√√

∞∏

k = −∞,

k = n

(λ2k−1 − ξn)(λ2k − ξn)

(ξk − ξn)2. (28)

So, expressions (21) and (28) imply (7), and the proof is concluded. ��

Notice that if, instead of the Dirichlet boundary conditions, we take theperiodic or the antiperiodic boundary conditions, then Eq. 21 remains λn = 0.Hence, the eigenvalues λn, n ∈ Z , of the periodic and antiperiodic problemsdo not depend on the parameter t.

5 Corollary and Remarks

Corollary 1 If instead of p(x, t) and q(x, t) we consider p(x + τ, t) and q(x +τ, t) then, as we have seen in the previous section, the eigenvalues of the periodicand antiperiodic problem do not depend on the parameters τ and t. However,the eigenvalues ξn of the Dirichlet problem and the signs σn depend on τ andt: ξn = ξn (τ, t), σn = σn(τ, t) = ±1, n ∈ Z . In this case, the system (7) takes theform

∂ξn

∂t= −2σn(τ, t)

√(ξn − λ2n−1)(λ2n − ξn) ·

√√√√√√

∞∏

k = −∞,

k = n

(λ2k−1 − ξn)(λ2k − ξn)

(ξk − ξn)2

×

⎧⎪⎨

⎪⎩q2(τ, t) + qx(τ, t) + [

p(τ, t) + ξn]2 + ξ 2

n − b(t)[

p(τ, t) + ξn]

+∞∑

k=−∞

αk(t)s1(π, λk, t, τ )

ξn − λk+

∞∫

−∞

β(λ, t)s1(π, λ, t, τ )

ξn − λdλ

⎫⎪⎬

⎪⎭. (29)

Here

s1(π, λ, t, τ ) = π

∞∏

k=−∞

ξk − λ

ak.


Using the trace formulas

p(τ, t) =∞∑

k=−∞

(λ2k−1 + λ2k

2− ξk(τ, t)

), (30)

and

q2(τ, t) + qx(τ, t) =∞∑

k=−∞

(λ2

2k−1 + λ22k

2− ξ 2

k (τ, t)

),

system (29) can be rewritten in closed form:

∂ξn

∂t= −σn(τ, t)

√(ξn − λ2n−1)(λ2n − ξn) ·

√√√√√√

∞∏

k = −∞,

k = n

(λ2k−1 − ξn)(λ2k − ξn)

(ξk − ξn)2

×

⎧⎪⎨

⎪⎩λ2

2n−1 + λ22n +

∞∑

k = −∞,

k = n

(λ2

2k−1 + λ22k − 2ξ 2

k

) + 1

2

×

⎡

⎢⎢⎢⎣λ2n−1 + λ2n +∞∑

k = −∞,

k = n

(λ2k−1 + λ2k − 2ξk)

⎤

⎥⎥⎥⎦

2

− b(t)

⎡

⎢⎢⎢⎣λ2n−1 + λ2n +∞∑

k = −∞,

k = n

(λ2k−1 + λ2k − 2ξk)

⎤

⎥⎥⎥⎦

+ 2∞∑

k=−∞


ξn − λk+2

∞∫

−∞

β(λ, t)s1(π, λ, t, τ )

ξn − λdλ

⎫⎬

⎭ , n ∈ Z .

(31)

Corollary 2 This theorem provides a method for solving the problem (1)–(4).Indeed, we denote the spectral data of the problem (4) corresponding to thecoef f icients p(x + τ, t) and q(x + τ, t) by λn, ξn(τ, t) and σn(τ, t), n ∈ Z .

The steps are the following: First, we f ind the spectral data λn, ξ 0n (τ ), and

σ 0n (τ ), n ∈ Z , corresponding to the coef f icients p0(x + τ) and q0(x + τ). Next

we solve the Cauchy problem

ξn(τ, t)|t=0 = ξ 0n (τ ), σn(τ, t)|t=0 = σ 0

n (τ ), n ∈ Z ,


for the system of Dubrovin–Trubowitz (31). After that, by using the formulas oftraces (30) and

q(τ, t) =∞∑

n=−∞σn(τ, t)

√(ξn − λ2n−1)(λ2n − ξn) ·

√√√√√√

∞∏

k = −∞k = n

(λ2k−1 − ξn)(λ2k − ξn)

(ξk − ξn)2,

we obtain the expressions of p(x, t) and q(x, t). With their help, we construct asolution of problem (1)–(4): u(x, t) = −p(x, t) + iq(x, t). Then it is easy to f indthe Floquet’s solution ψ(x, t, λ).

Corollary 3 If the number of zones is f inite, that is, there are two nonnegativeinteger numbers N and M such that λ2k−1 = λ2k = ξk for all k > N and k <

−M, then the system (31) takes the form

∂ξn

∂t= −σn(τ, t)

√(ξn − λ2n−1)(λ2n − ξn) ·

√√√√√√

N∏

k = −M,

k = n

(λ2k−1 − ξn)(λ2k − ξn)

(ξk − ξn)2

×

⎧⎪⎨

⎪⎩λ2

2n−1 + λ22n +

N∑

k = −M,

k = n

(λ2

2k−1 + λ22k − 2ξ 2

k

)

+ 1

2

⎡

⎢⎢⎢⎣λ2n−1 + λ2n +N∑

k = −M,

k = n

(λ2k−1 + λ2k − 2ξk)

⎤

⎥⎥⎥⎦

2

− b(t)

⎡

⎢⎢⎢⎣λ2n−1 + λ2n +N∑

k = −M,

k = n

(λ2k−1 + λ2k − 2ξk)

⎤

⎥⎥⎥⎦

+ 2∞∑

k=−∞


ξn − λk+ 2

∞∫

−∞

β(λ, t)s1(π, λ, t, τ )

ξn − λdλ

⎫⎬

⎭ ,

n = −M, ..., N.


Remark 1 Spectral parameters ξn(τ, t) and σn(τ, t), n ∈ Z also satisfy thefollowing system:

∂ξn

∂τ= σn(τ, t)

√(ξn − λ2n−1)(λ2n − ξn) ·

√√√√√√

∞∏

k = −∞,

k = n

(λ2k−1 − ξn)(λ2k − ξn)

(ξk − ξn)2

×

⎧⎪⎪⎪⎨

⎪⎪⎪⎩λ2n−1 + λ2n +

∞∑

k = −∞,

k = n

(λ2k−1 + λ2k − 2ξk)

⎫⎪⎪⎪⎬

⎪⎪⎪⎭, n ∈ Z .

Remark 2 Using the results obtained in [36] we can conclude that, if p0(x) andq0(x) are real analytical functions, then the p(x, t) and q(x, t) are also realanalytical functions on x.

Remark 3 In [35] an analogue of the inverse theorem of G.Borg for Dirac’soperator (6) is proven. According to this result we can conclude, if p0(x) andq0(x) are π

2 -periodic functions, then the p(x, t) and q(x, t) are also π2 -periodic

functions on x.

Remark 4 We can consider Eq. 1 coupled with the following more generalform of source

G(x, t) =∞∫

−∞iβ(λ, t)(ψ1 − iψ2)

(ψ1 − iψ2)

dλ (γ (λ, t)),

which contains above considered source.

Acknowledgements The author expresses his gratitude to Prof. Aknazar Khasanov (UrgenchState University, Uzbekistan) and Prof. Alberto Cabada (University of Santiago de Compostela,Spain), for a discussion and valuable advice, as well as to the Erasmus Mundus EC Lot 9 projectfor a postdoc grant at the University of Santiago de Compostela. The author also is grateful forthe suggestions of the anonymous referee which have improved the quality of this paper.

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The Neumann Type Systems and Algebro-GeometricSolutions of a System of Coupled Integrable Equations

Jinbing Chen · Zhijun Qiao

Received: 17 May 2009 / Accepted: 3 March 2011 / Published online: 24 March 2011© Springer Science+Business Media B.V. 2011

Abstract A system of (1+1)-dimensional coupled integrable equations is de-composed into a pair of new Neumann type systems that separate the spatialand temporal variables for this system over a symplectic submanifold. Then,the Neumann type flows associated with the coupled integrable equations areintegrated on the complex tour of a Riemann surface. Finally, the algebro-geometric solutions expressed by Riemann theta functions of the system ofcoupled integrable equations are obtained by means of the Jacobi inversion.

Keywords Integrable equations · Neumann type systems ·Algebro-geometric solutions

Mathematics Subject Classifications (2010) 37K10 · 37J35 · 70H06

1 Introduction

The Neumann system of harmonic oscillator constrained on the unit sphere is aprototype of finite dimensional integrable system (FDIS) with rich mathemati-cal natures in the area of classical mechanics [22]. Based on the Flaschka’s idea,Moser’s, Veselov’s and Knoerrer’s work [14, 19, 23, 24, 35], a number of new

J. ChenDepartment of Mathematics, Southeast University, Nanjing, Jiangsu 210096,People’s Republic of Chinae-mail: [email protected]

Z. Qiao (B)Department of Mathematics, University of Texas—Pan American, Edinburg,TX 78539, USAe-mail: [email protected]

172 J. Chen, Z. Qiao

FDISs of both Neumann and Bargmann types were found under a symmetricconstraint between spectral potentials and eigenfunctions in the frameworkof the nonlinearization of Lax pair [4, 5]. The FDISs of Bargmann type arethe canonical Hamiltonian systems produced under a Bargmann constraintfrom the Lax pair of an integrable equation; while the FDISs of Neumanntype are generated under a Neumann constraint on the symplectic submanifold[6, 9, 11, 27, 28, 33, 37, 38]. Those resultant FDISs not only enrich the contentof integrable systems itself, but also pave an effective way to solve integrableequations via the separation of spatial and temporal variables. It is alreadynoticed that finite dimensional integrable Hamiltonian systems have been usedto get algebro-geometric solutions through the finite parametric (or involutive)solutions of integrable equations with the help of the theory of algebraiccurves [1, 7, 16, 17, 28, 30, 31, 36, 37]. In particular, a Neumann type systemwas already applied by Qiao to obtain the algebro-geometric solution of theCamassa–Holm (CH) equation on a symplectic submanifold [33], where theLax matrix, dynamical r-matrix and Jacobi inversion were involved in.

To understand deeply the physical applications of integrable dynamicalsystems , one has to derive all kinds of explicit solutions for nonlinear evolutionequations from different standpoints. After the breakthrough discovery ofinverse scattering transformation [15], many interesting explicit solutions havebeen found, including the classical soliton solutions, the algebro-geometric(or finite-gap, quasi-periodic) solutions, and the polar expansion solutions.One can easily see that all explicit solutions of physical interests have a finitenumber of parameters. A deeper insight indicates that they may satisfy certainsolvable ordinary differential equations and can be obtained through tacklingthe associated FDISs, which are reduced from integrable equations. Apartfrom the fruitful application of finite dimensional integrable Hamiltoniansystems [1, 7, 16, 17, 28, 30, 31, 36, 37] and the work of the CH Neumannsystem with algebro-geometric solution [33], we also found that the Neumanntype flow is in essential the Hamiltonian flow in the sense of Dirac–Poissonbracket over a symplectic submanifold, and the Neumann constraint under thescheme of nonlinearization of Lax pair directly cast in a finite dimensionalinvariant submanifold in quite a few cases [11, 28, 33]. In particular, thegenerating function of integrals of motion of Neumann type system determinesa Riemann surface of hyperelliptic curve that pave a bridge to construct Abel–Jacobi (or angel) variables for integrable equations [12, 33]. Following theabove-mentioned analysis, in this paper we present a distinct way by using theNeumann type systems to derive new algebro-geometric solutions for moreintegrable equations of physical and mathematical interests.

To illustrate our scheme, we study the algebro-geometric solutions of thefollowing (1+1)-dimensional nonlinear evolution equations [34]{

ut = v−2vxvxx − v−1vxxx − 2uux − 4vvx,

vt = −2uvx − uxv.(1)

In fact, the system (1) is the coupled integrable equations from the TDhierarchy, which allows the zero-curvature representation in the sense of Lax

Neumann Type Systems and Algebro-Geometric Solutions 173

compatibility [20], the Hamiltonian structure in view of the trace identity [34],and the one- and two-soliton solutions by the Darboux transformation [10]. Inthe following, we will provide a feasible relation between two Neumann typesystems stemmed from the Lax pair of (1) and algebro-geometric solutionsof the integrable system (1). To see this, the integrable system (1) is reducedto two FDISs of Neumann type, whose compatible solutions yield solutionsof (1) through a direct algebraic operation [8]. An interesting thing is that twoNeumann type systems share the common Lax matrix and a dynamical r-matrixstructure in the Dirac–Poisson bracket [28, 32, 37, 39], instead of the standardPoisson bracket since we construct Neumann type systems on a symplecticsubmanifold.

The Lax matrix and the dynamical r-matrix guarantee that the twoNeumann type systems are completely integrable in the Liouville sense. Re-ferring to the approach for getting algebro-geometric solutions for (1+1)- and(2+1)-dimensional integrable equations [3, 7, 16, 17, 21, 28, 30, 31, 36, 37], twosets of elliptic variables are singled out from the entries of Lax matrix, andsolutions of the integrable system (1) are expressed by the symmetric func-tions with respect to these elliptic variables. Furthermore, through discussingthe Jacobi inversion, we attain the algebro-geometric solutions of integrablesystem (1) in terms of Riemann theta functions.

The whole paper is organized as follows. In the next section, we decomposethe integrable system (1) into two FDISs of Neumann type. In Section 3, theNeumann type flows are linearized/straightened out on the complex tour of aRiemann surface, and in Section 4 we derive the algebro-geometric solutionsof integrable system (1) through the Jacobi inversion.

2 Decomposition of Integrable Equations

To describe our results, we first collect some necessary notations and formulas.Let us begin with the spectral problem [34]

ϕx = Uϕ, U =

⎛⎜⎜⎝

−1

2λ + 1

2u −v

v1

2λ − 1

2u

⎞⎟⎟⎠ , ϕ =

(ϕ1

ϕ2

), (2)

where λ is a spectral parameter, and u and v are two spectral potentials. Inorder to derive the integrable hierarchy associated with (2), we define theLenard sequence {g j} (−1 � j ∈ Z) by

Kg j−1 = Jg j, Jg−1 = 0, j � 0, (3)

with

K =⎛⎜⎝−1

2∂v−1∂v−1∂ − 2∂ −1

2∂v−1u

−1

2uv−1∂ −1

2∂

⎞⎟⎠ , J =

⎛⎜⎝ 0 −1

2∂v−1

−1

2v−1∂ 0

⎞⎟⎠ , (4)


where ∂ = ∂/∂x and ∂−1 is the inverse of ∂ : ∂−1∂ = ∂∂−1 = 1. Noticing that thekernel of J is of dimension 2 with two generators g−1 = (0, 2v)T and g−2 =(

12 , 0

)T, one can easily get

ker J = {�1g−1 + �2g−2|∀�1, �2 ∈ R}.Each g j can be determined by the recursion formula (3). In particular, we have

g0 = (v2, 2uv

)T, g1 = (

2uv2, 2vxx + 2u2v + 4v3)T

. (5)

Let us consider an auxiliary spectral problem that is the time-dependent partof (2)

ϕtn = V(n)ϕ, V(n) =(

V(n)11 V(n)

12

V(n)21 −V(n)

11

), n � 1, (6)

where

V(n)11 = −1

4v−1∂v−1∂g(1) + 1

4(λ − u)v−1g(2), V(n)

12 = −1

2v−1∂g(1) + 1

2g(2),

V(n)21 = −1

2v−1∂g(1) − 1

2g(2), g = (

g(1), g(2))T =

n∑j=0

g j−2λn− j.

Then the compatibility condition of (2) and (6) gives the integrable hierarchy[34]

(u, v)Ttn = Jgn−1, n � 1. (7)

Apparently, the first nontrivial member of (7) is the integrable system (1) witht = t2, which is the compatibility condition of Lax pair (2) and

ϕt = V(2)ϕ, V(2) =⎛⎜⎝

1

2λ2 − 1

2u2 − 1

2v−1vxx λv − vx + uv

−λv − vx − uv −1

2λ2 + 1

2u2 + 1

2v−1vxx

⎞⎟⎠ .

(8)

In what follows, we want to decompose (1) into two Neumann type systemson a symplectic submanifold. Let us consider N copies of the spectral problem(2) with N distinct eigenvalues λ1, λ2, · · · , λN and their corresponding eigen-functions ϕ = (pj, q j)

T ,

(pj

q j

)x

=⎛⎜⎝−1

2λ j + 1

2u −v

v1

2λ j − 1

2u

⎞⎟⎠

(pj

q j

), 1 � j � N. (9)

One can readily calculate the functional gradient of each eigenvalue λ j withrespect to the spectral potentials u and v [9]

∇λ j = (δλ j/δu, δλ j/δv

)T =(

pjq j, −(

p2j + q2

j

))T. (10)


Taking into account the Neumann constraint [4, 5, 9]

g−1 =N∑

j=1

∇λ j, (11)

leads to

〈p, q〉 = 0, 〈p, p〉 − 〈q, q〉 = 0,

u = 〈�p, p〉 + 〈�q, q〉〈p, p〉 + 〈q, q〉 = 1

2

( 〈�p, p〉〈p, p〉 + 〈�q, q〉

〈q, q〉)

,

v = −〈p, p〉 + 〈q, q〉2

= −〈p, p〉, (12)

where p = (p1, · · · , pN)T , q = (q1, · · · , qN)T , � = diag(λ1, · · · , λN), and 〈·, ·〉stands for the standard inner product in R

N . In accordance with the rule ofthe nonlinearization of Lax pair, substituting (12) into (9) gives rise to the firstnonlinear dynamical system of Neumann type,⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

px = −1

2�p + 1

4

( 〈�p, p〉〈p, p〉 + 〈�q, q〉

〈q, q〉)

p + 〈p, p〉q,

qx = 1

2�q − 1

4

( 〈�p, p〉〈p, p〉 + 〈�q, q〉

〈q, q〉)

q − 〈q, q〉p,

〈p, q〉 = 0, 〈p, p〉 − 〈q, q〉 = 0.

(13)

On condition that the independent temporal variable t is regarded as theequivalence to the spatial variable x in the view point of mathematics, imposingthe Neumann constraint (12) onto the time-dependent part (8) leads to anothernew Neumann type system⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

pt = 1

2�2 p+ 〈�p, q〉p− 1

4

( 〈�2 p, p〉〈p, p〉 + 〈�2q, q〉

〈q, q〉)

p− 〈p, p〉�q− 〈�p, p〉q,

qt = 〈q, q〉�p + 〈�q, q〉p − 1

2�2q − 〈�p, q〉q + 1

4

( 〈�2 p, p〉〈p, p〉 + 〈�2q, q〉

〈q, q〉)

q,

〈p, q〉 = 0, 〈p, p〉 − 〈q, q〉 = 0.

(14)

A direct but lengthy computation yields the following proposition

Proposition 1 Let (p(x, t), q(x, t))T be the compatible solution of the twoNeumann type systems (13) and (14), then

u(x, t) = 1

2

( 〈�p, p〉〈p, p〉 + 〈�q, q〉

〈q, q〉)

, v(x, t) = −〈p, p〉, (15)

are solutions of the integrable equations (1).


So, by this proposition, the integrable equations (1) can be solved with afinite parametric solution (15) through solving a pair of (finite dimensional)nonlinear dynamical systems of ordinary differential equations (13) and (14).

By using the procedure shown in [9, 28, 31, 32, 37, 39], we know that theNeumann type system (13) admits the Lax representation

Lx(λ) = [U, L(λ)], Lx(λ) = ∂L(λ)/∂x, (16)

where

L(λ) =⎛⎜⎝

1

20

0 −1

2

⎞⎟⎠ +

N∑j=1

1

λ − λ j

(q j p j −p2

jq2

j −q j p j

)�

(A(λ) B(λ)

C(λ) −A(λ)

), (17)

and

U =

⎛⎜⎜⎜⎝

−1

2λ + 1

4

( 〈�p, p〉〈p, p〉 + 〈�q, q〉

〈q, q〉)

〈p, p〉

−〈p, p〉 1

2λ − 1

4

( 〈�p, p〉〈p, p〉 + 〈�q, q〉

〈q, q〉)

⎞⎟⎟⎟⎠ .

(18)

Actually, the Lax matrix (17) was first discussed in [28, 29, 32, 39] to classifythe FDISs. A very interesting fact is that the Neumann type system (14),i.e. the nonlinearization of the time-dependent part (8) under the Neumannconstraint, admits the Lax representation with the same Lax matrix L(λ)

defined by (17)

Lt(λ) = [V(2), L(λ)], Lt(λ) = ∂L(λ)/∂t, (19)

where

V(2) =(

V(2)11 −λ〈p, p〉 − 〈�p, p〉

λ〈q, q〉 + 〈�q, q〉 −V(2)11

), (20)

with

V(2)11 = 1

2λ2 + 〈�p, q〉 − 1

4

( 〈�2 p, p〉〈p, p〉 + 〈�2q, q〉

〈q, q〉)

.

The Neuamnn type systems (13) and (14) are completely integrable in theLiouville sense since L(λ) satisfies a dynamical r-matrix structure in the Dirac–Poisson bracket [9, 32, 38, 39]. Consequently, this assures the compatibility ofthe two Neumann type systems (13) and (14), which implies that the Neumanntype flows mutually commute [2].

3 Straightening Out of the Neumann Type Flows

To get explicit solutions of integrable system (1), we adopt the procedureof straightening out Neumann type flows that are restricted on a symplectic


submanifold. To do this, we select two sets of elliptic variables μ1, μ2, · · · ,

μN−1 and ν1, ν2, · · · , νN−1 from the entries of L(λ),

B(λ) = −N∑

j=1

p2j

λ − λ j= −〈p, p〉m(λ)

a(λ),

C(λ) =N∑

j=1

q2j

λ − λ j= 〈q, q〉n(λ)

a(λ),

(21)

where

a(λ) =N∏

k=1

(λ − λk), m(λ) =N−1∏k=1

(λ − μk), n(λ) =N−1∏k=1

(λ − νk). (22)

The combination of (21) and (22) gives

〈�p, p〉〈p, p〉 =

N∑j=1

λ j −N−1∑j=1

μ j � σ − σ1,

〈�q, q〉〈q, q〉 =

N∑j=1

λ j −N−1∑j=1

ν j � σ − σ2. (23)

By (12) and (20), one obtains

u = σ − 1

2(σ1 + σ2), ∂x ln v = 1

2(σ1 − σ2), (24)

and{

V(2)12 = −〈p, p〉(λ + σ − σ1),

V(2)21 = 〈q, q〉(λ + σ − σ2).

(25)

Define

det L(λ) = −A(λ)2 − B(λ)C(λ) = − b(λ)

4a(λ)= − R(λ)

4a2(λ), (26)

where

b(λ) =N∏

k=1

(λ − λN+k), R(λ) = a(λ)b(λ) =2N∏k=1

(λ − λk).

It follows from (21), (22) and (26) that

A(μk) =√

R(μk)

2a(μk), A(νk) =

√R(νk)

2a(νk), 1 � k � N − 1. (27)


By (21), (16) and (19), we arrive at the evolution equation of all μk and νk

regarding x and t,

dμk

dx= −

√R(μk)

N−1∏i=1,i �=k

(μk − μi)

,dνk

dx=

√R(νk)

N−1∏i=1,i �=k

(νk − νi)

, 1 � k � N − 1,

(28)

and ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

dμk

dt= (μk − σ1 + σ)

√R(μk)

N−1∏i=1,i �=k

(μk − μi)

,

dνk

dt= (−νk + σ2 − σ)

√R(νk)

N−1∏i=1,i �=k

(νk − νi)

,

1 � k � N − 1. (29)

These formulas naturally lead to the consideration of the Riemann surface

of hyperelliptic curve given by the equation ξ 2 = R(λ), whose genus is N − 1.For the same λ, there exist two points (λ,

√R(λ)) and (λ, −√

R(λ)) on theupper and lower sheets of , and there are two points at infinity that are not thebranch points because degR(λ) = 2N. Under an alternative local coordinatez = λ−1, they are marked as ∞1 = (0, 1) and ∞2 = (0, −1).

Let a1, a2, · · · , aN−1; b 1, b 2, · · · , b N−1 be a set of regular cycle paths on ,which are automatically independent if they have the intersection numbers

ai ◦ a j = bi ◦ b j = 0, ai ◦ b j = δij, i, j = 1, 2, · · · , N − 1.

It is well known that

ωl = λl−1dλ√R(λ)

, 1 � l � N − 1,

are N − 1 linearly independent holomorphic differentials of . Let

Aij =∫

a j

ωi, C = (Aij)−1, 1 � i, j � N − 1, (30)

then ωl can be normalized into a new basis ω j,

ω j =N−1∑l=1

C jlωl,

∫ai

ω j =N−1∑l=1

C jl

∫ai

ωl =N−1∑l=1

C jl Ali = δ ji,

and each

Bij =∫

b j

ωi, 1 � i, j � N − 1,


is an entry of (N − 1) × (N − 1) matrix B = (Bij) that characterizes theRiemann surface and applies to construct Riemann theta functions of .Let p0 be a fixed point, then the Abel–Jacobi variables can be given by

ρ(1)

j (x, t) =N−1∑k=1

∫ μk(x,t)

p0

ω j =N−1∑k=1

N−1∑l=1

C jl

∫ μk

p0

λl−1dλ√R(λ)

,

ρ(2)

j (x, t) =N−1∑k=1

∫ νk(x,t)

p0

ω j =N−1∑k=1

N−1∑l=1

C jl

∫ νk

p0

λl−1dλ√R(λ)

,

1 � j � N − 1.

(31)

Taking derivative with respect to x on both sides of (31)1 leads to

∂xρ(1)

j =N−1∑l=1

N−1∑k=1

C jlμl−1

k μk,x√R(μk)

=N−1∑l=1

N−1∑k=1

C jl−μl−1

kN−1∏

i=1,i �=k(μk − μi)

. (32)

With the help of the formulae [26],

Is =N−1∑k=1

μsk

N−1∏i=1,i �=k

(μk − μi)

= δs,N−2, IN−1 = σ1 IN−2, 1 � s � N − 2,

(33)we obtain

∂xρ(1)

j = �(0)

j , �(0)

j = −C jN−1, 1 � j � N − 1. (34)

A similar calculation directly yields

∂tρ(1)

j = �(1)

j , ∂xρ(2)

j = −�(0)

j , ∂tρ(2)

j = −�(1)

j , (35)

where �(1)

j = C jN−2 + σC jN−1. Clearly, ρ(1)

j and ρ(2)

j can be integrated andwritten as linear superpositions in the flow variables x and t,

ρ(1)

j = �(0)

j x + �(1)

j t + γ(1)

j ,

ρ(2)

j = −�(0)

j x − �(1)

j t + γ(2)

j ,1 � j � N − 1, (36)

where

γ(1)

j =N−1∑k=1

∫ μk(0,0)

p0

ω j, γ(2)

j =N−1∑k=1

∫ νk(0,0)

p0

ω j,

are two integral constants.

4 Algebro-Geometric Solutions of the Integrable Equations

Since the Abel–Jacobi solutions (ρ(1), ρ(2)) (see (36)) are solved explicitly,the remaining steps are to write down the explicit expression of u and v of


integrable system (1). For this purpose, we turn to the procedure of Jacobiinversion

(ρ(1), ρ(2)) =⇒ (μk, νk).

Let T be the lattice in CN−1, which is generated by 2(N − 1) periodic vectors

{δi, B j}. Then we have the following complex tour—called Jacobian J() =C

N−1/T of . The Abel map is defined by

A : Div() → J(), A(p) =(∫ p

p0

ω1, · · · ,

∫ p

p0

ωN−1

),

where p is an arbitrary point on . Moreover, A can linearly be extended tothe factor group

Div() : A(∑

nk pk

)=

∑nkA( pk).

From [18, 25], the Riemann theta function is defined by

θ(ζ ) =∑

z∈ZN−1

exp (π i〈Bz, z〉 + 2π i〈ζ, z〉), ζ ∈ CN−1,

〈Bz, z〉 =N−1∑i, j=1

Bijziz j, 〈ζ, z〉 =N−1∑i=1

ziζi.

Let us consider two special divisors∑N−1

k=1 p(m)

k ,

A(

N−1∑k=1

p(m)

k

)=

N−1∑k=1

A(

p(m)

k

)=

N−1∑k=1

∫ p(m)

k

p0

ω = ρ(m), m = 1, 2,

where p(1)

k = (μk, ζ(μk)) and p(2)

k = (νk, ζ(νk)). Conforming to the Riemanntheorem [18], there exist two constant vectors (called Riemann constants)M(1), M(2) ∈ C

N−1 determined by such that

• f (1)(λ) � θ(A(ζ(λ)) − ρ(1) − M(1)) has N − 1 simple zeros at μ1, · · · ,

μN−1,• f (2)(λ) � θ(A(ζ(λ)) − ρ(2) − M(2)) has N − 1 simple zeros at ν1, · · · , νN−1.

To make the functions single valued, is cut by all paths ak, b k to form a simplyconnected region whose boundary is denoted by γ . By the residue formulas,one gets

N−1∑j=1

μ j = I() −2∑

s=1

Resλ=∞s

λd ln f (1)(λ),

N−1∑j=1

ν j = I() −2∑

s=1

Resλ=∞s

λd ln f (2)(λ), (37)


where

I() = 1

2π i

∮γ

λd ln f (m)(λ) =N−1∑j=1

∫a j

λω j, m = 1, 2,

is a constant independent of ρ(m) [13, 36]. The only requirement is to calculatethe residues at both infinities:

f (m)(λ)|λ=∞s = θ

(∫ p

p0

ω − ρ(m) − M(m)

)= θ

(∫ p

∞s

ω − πs − ρ(m) − M(m)

)

= θ

(· · · ,

∫ p

∞s

ω j − πsj − ρ(m)

j − M(m)

j , · · ·)

= θ

(· · · , ρ

(m)

j + M(m)

j + πsj + (−1)s

×(

C jN−1z + 1

2

(C jN−2 + σC jN−1

)z2 + · · ·

), · · ·

)

= θ(m)s

(ρ(m) + M(m) + πs

) + (−1)s+mθ(m)s,x z + · · · ,

where πsj = ∫ p0

∞sω j (s, m = 1, 2). Therefore, we arrive at

Resλ=∞s

λd ln f (m)(λ) = (−1)s+m∂x ln θ(m)s , (38)

where

θ(1)s = θ(�(0)x + �(1)t + ϒs), θ (2)

s = θ(−�(0)x − �(1)t + �s),

with

ϒsj = γ(1)

j + M(1)

j + πsj, �sj = γ(2)

j + M(2)

j + πsj, 1 � j � N − 1.

From (37) and (38), we have

N−1∑l=1

μl = I() + ∂x lnθ

(1)2

θ(1)1

,

N−1∑l=1

νl = I() + ∂x lnθ

(2)1

θ(2)2

. (39)

Substituting (39) into (24), we get the algebro-geometric solutions of integrablesystem (1),

u = −1

2∂x ln

θ(�(0)x + �(1)t + ϒ2)

θ(�(0)x + �(1)t + ϒ1)

θ(−�(0)x − �(1)t + �1)

θ(−�(0)x − �(1)t + �2)− I() + σ,

v2= θ(�(0)x+�(1)t+ϒ2)

θ(�(0)x+�(1)t+ϒ1)

θ(−�(0)x−�(1)t+�2)

θ(−�(0)x−�(1)t+�1)

θ(�(1)t+ϒ1)

θ(�(1)t+ϒ2)

θ(−�(1)t+�1)

θ(−�(1)t+�2)v2(0,t).

In conclusion, the algebro-geometric solutions of integrable system (1) areattained, which implies that the two Neumann type systems in this paper aresuccessfully used to derive algebro-geometric solutions of integrable equationsin (1+1)-dimensional just like the procedure shown in [33]. This procedure


is different from the utilization of finite dimensional integrable Hamiltoniansystems in the case of Bargmann constraint [19, 24, 35] that corresponds to thewhole symplectic space. We will try to solve some other integrable equationsunder the Neumann constraint.

Acknowledgements The authors greatly appreciate the referee for his/her helpful suggestionsand comments.

Chen is supported by the National Natural Science Foundation of China (Grant No. 11001050),and Qiao by the U. S. Army Research Office under contract/grant number W911NF-08-1-0511 andthe Texas Norman Hackerman Advanced Research Program under Grant 003599- 0001-2009.

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Spinorial Characterizations of Surfacesinto 3-dimensional Pseudo-Riemannian Space Forms

Marie-Amélie Lawn · Julien Roth

Received: 27 May 2010 / Accepted: 13 May 2011 / Published online: 7 June 2011© Springer Science+Business Media B.V. 2011

Abstract We give a spinorial characterization of isometrically immersed sur-faces of arbitrary signature into 3-dimensional pseudo-Riemannian spaceforms. This generalizes a recent work of the first author for spacelike immersedLorentzian surfaces in R

2,1 to other Lorentzian space forms. We also charac-terize immersions of Riemannian surfaces in these spaces. From this we candeduce analogous results for timelike immersions of Lorentzian surfaces inspace forms of corresponding signature, as well as for spacelike and timelikeimmersions of surfaces of signature (0, 2), hence achieving a complete spinorialdescription for this class of pseudo-Riemannian immersions.

Keywords Dirac operator · Killing spinors · Isometric immersions ·Gauss and Codazzi equations

Mathematics Subject Classifications (2010) 53C27 · 53B25 · 53B30 · 53C80

M.-A. LawnInstitut de Mathématiques, Université de Neuchâtel,Rue Emile-Argand 11, 2000, Neuchâtel, Suissee-mail: [email protected]

J. Roth (B)Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Université Paris-EstMarne-la-Vallée, Cité Descartes, Bâtiment Copernic, Bureau 4B097, 5, Boulevard Descartes,Champs sur Marne, 77454 Marne-la-Vallée, Cedex 2, Francee-mail: [email protected]

186 M.-A. Lawn, J. Roth

1 Introduction

A fundamental question in the theory of submanifolds is to know whether a(pseudo-)Riemannian manifold (Mp,q, g) can be isometrically immersed intoa fixed ambient manifold (M

r,s, g), where p (resp. r) stands for the positive, and

q (resp. s) for the negative eigenvalues of g (resp. g). In this paper, we focus onthe case of hypersurfaces (i.e of codimension 1), and more especially of sur-faces of arbitrary signature into 3-spaces. If the ambient space is a space form,as the pseudo-Euclidean space R

p,q and the pseudo-spheres Sp,q of positive

constant curvature, or the pseudo-hyperbolic spaces Hp,q of negative constant

curvature, the answer is given by the well-known fundamental theorem ofhypersurfaces:

Theorem [9] (Mp,q, g) be a pseudo-Riemannian manifold with signature(p, q), p + q = n. Let A be a symmetric Codazzi tensor (i.e d∇ A(X, Y) :=∇X A(Y) − ∇Y A(X) − A([X, Y]) = 0), satisfying

R(X, Y)Z = δ[ 〈A(Y), Z 〉 A(X) − 〈A(X), Z 〉 A(Y)

]

+ κ[ 〈Y, Z 〉 X − 〈X, Z 〉 Y

]

with κ ∈ R for all x ∈ M and X, Y, Z ∈ Tx M.Then, if δ = 1 (resp. δ = −1), there exists locally an isometric immersion of

M with spacelike (resp. timelike) normal vector ν, i.e with ‖ν‖ = 1 (resp. ‖ν‖ =−1), into M

p+1,q(κ) (resp. Mp,q+1(κ)).

In the Riemannian case and for small dimensions (n = 2 or 3), another nec-essary and sufficient condition is now well-known. This condition is expressedin spinorial terms, namely, by the existence of a special spinor field. This workinitiated by Friedrich [4] in the late 90’s for surfaces of R

3 was generalizedfor surfaces of S

3 and H3 [8] and other 3-dimensional homogeneous manifolds

[10].The first author [5] uses this approach to give a spinorial characterization

of space-like immersions of Lorentzian surfaces in the Minkowski spaceR

2,1. In this paper, we give a generalization of this result to Lorentzian andRiemannian surfaces into one of the three Lorentzian space forms, R

2,1, S2,1

or H2,1. This finally allows us to give a complete spinorial characterization for

spacelike as well as for timelike immersions of surfaces of arbitrary signatureinto pseudo-Riemannian space forms.

We will begin by a section of recalls about extrinsic pseudo-Riemannianspin geometry. For further details, one refers to [1, 2] for basic facts about spingeometry and [1, 3, 7] for the extrinsic aspect.

Spinorial Characterizations of Surfaces into 3-dimensional... 187

2 Preliminaries

2.1 Pseudo-Riemannian Spin Geometry

Let (Mp,q, g), p + q = 2, be an oriented and time-oriented pseudo-Riemannian surface of arbitrary signature isometrically immersed into a three-dimensional pseudo-Riemannian spin manifold (Nr,s, g). We introduce theparameter ε as follows: ε = i if the immersion is timelike and ε = 1 if theimmersion is spacelike. Let ν be a unit vector normal to M. The fact thatM is oriented implies that M carries a spin structure induced from the spinstructure of N and we have the following identification of the spinor bundlesand Clifford multiplications:

{�N|M ≡ �M.

X · ϕ|M = (εν • X • ϕ

)|M,

with X ∈ �(T M) and where · and • are the Clifford multiplications, respec-tively on M and N.

Moreover, we have the following well-known spinorial Gauss formula

∇Xϕ = ∇Xϕ − ε

2A(X) · ϕ, (1)

for X ∈ �(T M), with ∇ and ∇ respectively the spin connections on N and M,and where A is the shape operator of the immersion.

Finally we denote by R the spinorial curvature and we recall the Ricciidentity on M

R(e1, e2)ϕ = 12ε1ε2 R1221e1 · e2 · ϕ, (2)

where e1, e2 is a local orthonormal frame of M and ε j = g(e j, e j).The complex volume element on the surface depends on the signature and

is defined by

ωC

p,q = iq+1e1 · e2.

Obviously ωC

p,q2 = 1, independently of the signature, and the action of ωC splits

�M into two eigenspaces �±M of real dimension 2. Therefore, a spinor fieldϕ can be written as ϕ = ϕ+ + ϕ− with ωC · ϕ± = ±ϕ±. Finally, we denote ϕ =ωC · ϕ = ϕ+ − ϕ−.

2.2 Restricted Killing Spinors

Let (Mp,q, g), p + q = 2 be an oriented and time-oriented surface of thepseudo-Riemannian space form M

r,s(κ), r + s = 3, p � r, q � s. This spaceform carries a Killing spinor ϕ, that is satisfying ∇Xϕ = λX • ϕ, with κ = 4λ2,


X ∈ �(T N). From the Gauss formula (1), the restriction of ϕ on M satisfiesthe equation

∇Xϕ = ε

2A(X) · ϕ + λX • ϕ. (3)

for any vector field X tangent to M. But we have

X • ϕ = ε2ν • ν • X • ϕ = −ε2ν • X • ν • ϕ = −εX · (ν • ϕ).

Moreover, the complex volume element ωC

r,s = −ise1 • e2 • ν of Mr,s(κ) over M

acts as the identity on �Mr,s(κ)|M ≡ �M. Thus, we have

ν • ϕ = ωC

r,s • ν • ϕ = −isν • e1 • e2 • ν • ϕ

= isν • e1 • ν • e2 • ϕ

= isε2(εν • e1) • (ενe2 • ϕ)

= isε2e1 · e2 · ϕ.

Hence a simple case by case computation shows that we have

X • ϕ = −isε3 X · e1 · e2 · ϕ = iX · ωC

p,q · ϕ = iX · ϕ.

in the six possible cases (for the three possible signatures (2,0), (1,1), (0,2) ofthe surface with respectively ε = 1 or i) and finally (3) becomes

∇Xϕ = ε

2A(X) · ϕ + iλX · ϕ. (4)

We will call a spinor solution of (4) a real special Killing spinor (RSK)-spinor if ε ∈ R, and an imaginary special Killing spinor (ISK)-spinor if ε ∈ iR.Then, such a spinor field satisfies the following Dirac-type equation

Dϕ = −2εHϕ − 2iλϕ, (5)

where D : �M → �M, D = ∑p+qi εiei · ∇ei , with εi = g(ei, ei), is the Dirac

operator on the surface.

2.3 Norm Assumptions

In this section, we precise the norm assumptions. Let (Mp,q, g) be a pseudo-Riemannian surface and ϕ a spinor field on M. Let ε = 1 or i and λ ∈ R oriR. We say that ϕ satisfies the norm assumption N±(p, q, λ, ε) if the followingholds:

1. For p = 2, q = 0 or p = 0, q = 2:

• If ε = 1, then X|ϕ|2 = ±2e 〈iλX · ϕ, ϕ〉 .

• If ε = i, then X 〈ϕ, ϕ〉 = ±2e 〈iλX · ϕ, ϕ〉 .

2. For p = 1, q = 1: ϕ is non-isotropic.


These conditions are satisfied by restricted Killing spinors, they are a directconsequence of (4). Moreover, as we will see, they are necessary in additionof the Dirac equation (5) to get an isometric immersion in the appropriateambient space.

We use these notation N±(p, q, λ, ε) for a sake of clarity in the statement ofthe main result.

3 The Main Result

We now state the main result of the present paper.

Theorem 1 Let (Mp,q, g), p + q = 2 be an oriented and time-oriented pseudo-Riemannian manifold. Let H be a real-valued function. Then, the three follow-ing statements are equivalent:

1. There exist two nowhere vanishing spinor f ields ϕ1 and ϕ2 satisfying the normassumptions N−(p, q, λ, ε) and N+(p, q, λ, ε) respectively and

Dϕ1 = 2εHϕ1 + 2iλϕ1 and Dϕ2 = −2εHϕ2 − 2iλϕ2.

2. There exist two spinor f ields ϕ1 and ϕ2 satisfying

∇Xϕ1 = −ε

2A(X) · ϕ1 − iλX · ϕ1, and ∇Xϕ2 = ε

2A(X) · ϕ2 + iλX · ϕ2,

where A is a g-symmetric endomorphism and H = − 12 tr (A).

3. There exists a local isometric immersion from M into the (pseudo)-Riemannian space form M

p+1,q(4λ2) (resp. Mp,q+1(4λ2)) if ε = 1 (resp. ε = i)

with mean curvature H and shape operator A.

Remark 1 Note that, in this result, two spinor fields are needed to getan isometric immersion. However, for the case of Riemannian surfaces inRiemannian space forms (Friedrich [4] and Morel [8]) only one spinor solutionof one of the two equations is sufficient. This is also the case for surfaces ofsignature (0, 2) in space forms of signature (0, 3).

In order to prove this theorem, we give two technical lemmas.

Lemma 1 Let (Mp,q, g) be an oriented pseudo-Riemannian surface and λ acomplex number. If M carries a spinor f ield solution of the equation

Dϕ = ± (εHϕ + 2iλϕ) (6)

satisfying the norm assumption N±(p, q, λ, ε), then this spinor satisf ies

∇Xϕ = ±(ε

2A(X) · ϕ − iλX · ϕ

).

Proof Both cases for sign + and − are the same, so, we give only the proof forthe sign + .


Case of signature (1, 1) We define the endomorphism Bϕ by

(Bϕ)ij = g(Bϕ(ei), e j) = βϕ(ei, e j) := 〈ε∇eiϕ, e j · ϕ〉.

Since ϕ is non-isotropic, ei·ϕ±〈ϕ+,ϕ−〉 is a normalized dual frame of �∓M and by the

same proof as in [5] we can show that

〈∇Xϕ, ei · ϕ±〉 = 〈ε∇Xϕ, εei · ϕ±〉 = − 12ε〈ϕ+, ϕ−〉〈Bϕ(X) · ϕ, ei · ϕ∓〉.

and hence ∇Xϕ = − 12ε〈ϕ+,ϕ−〉 Bϕ(X) · ϕ. Moreover

βϕ(e1, e2) = 〈∇e1ϕ, e2 · ϕ〉 = −〈ε∇e1ϕ, e21 · e2 · ϕ〉

= −〈εe1 · ∇e1ϕ, e1 · e2 · ϕ〉 = −〈εDϕ + εe2 · ∇e2ϕ, e1 · e2 · ϕ〉= −ε2 H〈ϕ, e1 · e2 · ϕ〉 − 〈2iελωC · ϕ, e1 · e2 · ϕ〉 + βϕ(e2, e1)

= −〈2iελωC · ϕ, e1 · e2 · ϕ〉 + βϕ(e2, e1),

since for any ϕ, ψ ∈ �(�M)

〈ϕ, e1 · e2 · ψ〉 = 〈e2 · e1 · ϕ, ψ〉 = −〈e1 · e2 · ϕ, ψ〉 = −〈ϕ, e1 · e2 · ψ〉 = 0.

Let us now consider the decomposition βϕ(X, Y) = Sϕ(X, Y) + Tϕ(X, Y) inthe symmetric part Sϕ and antisymmetric part Tϕ . We see easily that if λ/ε ∈iR, then βϕ is symmetric, i.e., Tϕ = 0. and if λ/ε ∈ R, then Tϕ(X) = 2iλ/ε ωC ·X. In the two cases, we have

∇Xϕ = ε

2A(X) · ϕ − iλX · ϕ,

by setting A = 2Sϕ . We verify easily that tr(A) = 2tr(Sϕ) = 2tr(Bϕ) = −2H.

Case of signature (2, 0) or (0, 2) The proof is fairly standard following thetechnique used in [4, 8, 10]. We consider the tensors Q±

ϕ defined by

Q±ϕ (X, Y) = e

⟨ε∇Xϕ±, Y · ϕ∓⟩

.

Then, we have

tr (Q±ϕ ) = −e

⟨εDϕ±, ϕ∓⟩ = −e

⟨ε(εH ± 2iλϕ∓, ϕ∓⟩

= −ε2(H ± 2e(λ))|ϕ∓|2.

Moreover, we have the following defect of symmetry of Q±ϕ ,

Q±ϕ (e1, e2) = e

⟨ε∇e1ϕ

±, e2 · ϕ∓⟩ = e⟨εe1 · ∇e1ϕ

±, e1 · e2 · ϕ∓⟩

= e⟨εDϕ±, e1 · e2 · ϕ∓⟩ − e

⟨ε∇e2ϕ

±, e1 · e2 · ϕ∓⟩

= e⟨(ε2 H ± 2iελ)ϕ∓, e1 · e2 · ϕ∓⟩ + e

⟨ε∇e2ϕ

±, e1 · ϕ∓⟩

= 2e(ελ)|ϕ∓|2 + Q±ϕ (e2, e1).


Then, using the fact that εe1 · ϕ±|ϕ±|2 and εe2 · ϕ±

|ϕ±|2 form a local orthonormalframe of �∓M for the real scalar product e 〈·, ·〉, we see easily that

∇Xϕ+ = εQ+

ϕ (X)

|ϕ−|2 · ϕ− and ∇Xϕ− = εQ−

ϕ (X)

|ϕ+|2 · ϕ+.

We set W = Q+ϕ

|ϕ−|2 − Q−ϕ

|ϕ+|2 . From the above computations, we have immediatelythat W + e (iλ/ε) Id is symmetric and trace-free. Now, we will show that W +e (iλ/ε) Id is of rank at most 1. First, we have

X|ϕ+|2 + ε2 X|ϕ−|2 = 2e⟨εW(X) · ϕ−, ϕ+⟩

.

Moreover, from the norm assuption N+(p, q, λ, ε), we have

X|ϕ+|2 + ε2 X|ϕ−|2 = 2e 〈iλX · ϕ, ϕ〉 = 4e⟨iλX · ϕ−, ϕ+⟩

.

We deduce immediately that W + 2e (iλ/ε) Id is of rank at most 1 and hencevanishes identically since it is symmetric and trace-free. Thus, we have thefollowing relation

|ϕ+|2 Q+ϕ − |ϕ−|2 Q−

ϕ = −2e(iλ/ε)|ϕ+|2|ϕ−|2g.

From now on, we will distinguish two cases.

• Case 1: iλ/ε ∈ R.

Then we are in one of these two possible situations: ε = i and λ ∈ R or ε = 1and λ ∈ iR. The second situation was studied by Morel [8].

So we define the following tensor F := Q+ϕ − Q−

ϕ + 2iελ(|ϕ+|2 − |ϕ−|2)g.We have then

∇Xϕ = ∇Xϕ+ + ∇Xϕ− = εQ+

ϕ (X)

|ϕ−|2 · ϕ− + εQ−

ϕ (X)

|ϕ+|2 · ϕ+

= εF(X)

|ϕ|2 · (ϕ+ + ϕ−) + iλX · ϕ− − iλX · ϕ+

= ε

2A(X) · ϕ − iλX · ϕ,

where we have set A = 2F|ϕ2| . We conclude by noticing that A is a symmetric

tensor with tr (A) = −2H.

• Case 2: iλ/ε ∈ iR.

Then we are in one of these two possible situations: ε = i and λ ∈ iR or ε = 1and λ ∈ R. The second situation was studied by Morel [8].


In this case, we have from the previous computations that W vanishesidentically. So we set

F = Q+ϕ

|ϕ−|2 = Q−ϕ

|ϕ+|2

and then we have ∇Xϕ = F(X) · ϕ, where F(X) is defined by g(F(X), Y) =F(X, Y). However, F is not symmetric. We define the following symmetrictensor A(X, Y) = 1

|ϕ|2 (F(X, Y) + F(Y, X)). We compute immediately

A(e1, e1) = 2F(e1, e1)/|ϕ|2 , A(e2, e2) = 2F(e2, e2)/|ϕ|2,

A(e1, e2) = 2F(e1, e1)/|ϕ|2 − 2λ/ε and A(e2, e2) = 2F(e2, e2)/|ϕ|2 + 2λ/ε.

Finally, we conclude that

∇Xϕ = ε

2A(X) · ϕ + λX · ω · ϕ = ε

2A(X) · ϕ − iλX · ϕ.

��

Lemma 2 Let (Mp,q, g) be an oriented pseudo-Riemannian surface and η, λ twocomplex numbers. If M carries a spinor f ield satisfying

∇Xϕ = ηA(X) · ϕ + iλX · ϕ,

then, we have

(−ε1ε2 R1212 + 4η2 det(A) + 4λ2)e1 · e2 · ϕ = 2ηd∇ A(e1, e2) · ϕ.

Proof An easy computation yields

∇X∇Yϕ = η∇X(A(Y)) · ϕ + η2 A(Y) · A(X) · ϕ + iηλA(Y) · X · ωC · ϕ

+ iλ∇XY · ωC · ϕ + iηλY · ωC · A(X) · ϕ − λ2Y · ωC · X · ωC · ϕ.

Hence (the other terms vanish by symmetry)

R(e1, e2)ϕ = ∇e1∇e2ϕ − ∇e2∇e1ϕ − ∇[e1,e2]ϕ

= η(∇e1 A(e2) − ∇e2 A(e1) − A([e1, e2])

)

· ϕ + η2(A(e2) · A(e1) − A(e1) · A(e2)) · ϕ

− λ2(e2 · ωC · e1 · ωC − e1 · ωC · e2 · ωC) · ϕ.

Since we have

A(e2) · A(e1) − A(e1) · A(e2) = −2 det(A)e1 · e2


and

e2 · ωC · e1 · ωC − e1 · ωC · e2 · ωC = e1 · e2 · (ωC)2 − e2 · e1 · (

ωC)2 = 2e1 · e2,

by Ricci identity (2), we get

12ε1ε2 R1221e1e2 · ϕ = ηd∇ A(e1, e2) − 2η2 det(A)e1 · e2ϕ − 2λ2e1 · e2 · ϕ,

and finally

(−ε1ε2 R1212 + 4η2 det(A) + 4λ2)e1 · e2 · ϕ = 2ηd∇ A(e1, e2) · ϕ. (7)

��

Now, we can give the proof of Theorem 1. We have already seen in thesection of preliminaries that 3 implies 2 which implies 1. Moreover, Lemma 1shows that 1 implies 2. Now, we will prove that 2 implies 3. For this, weuse Lemma 2, but we need to distinguish the three cases for the differentsignatures. Let ϕ = ϕ+ + ϕ−.

Case of signature (2, 0) Here, ωC = ie1e2, hence e1 · e2 · ϕ = −iωC · ϕ = −iϕ.Hence formula (7) becomes

−i (−R1212 + ε2 det(A) + 4λ2)︸︷︷︸G2,0

ϕ = ε d∇ A(e1, e2)︸︷︷︸C2,0

·ϕ.

or equivalently εC2,0 · ϕ± = ±iG2,0ϕ∓. Applying twice this relation we have

finally

ε2||C2,0||2ϕ± = −G22,0ϕ

±.

Again we have two cases.

• Spacelike immersion: ε = 1, M2,0 ↪→ M3,0.

We refer to [4] for the immersion in R3,0 and to [8] for S

3 and H3. Only one

(RSK)-spinor is needed.• Timelike immersion: ε = i, M2,0 ↪→ M

2,1.Two (ISK)-spinors are needed. We deduce from the above relationsbetween ϕ±

1 and ϕ±2 that

⟨C2,0 · ϕ1, ϕ2

⟩ = 0. Moreover, in this case we have〈ϕ1, ϕ2〉 = 0. Thus, since the spinor bundle �M is of complex rank 2, wehave C2,0 · ϕ1 = fϕ1 where f is a complex-valued function over M. By tak-ing the inner product by ϕ1, we see immediately that f only takes imaginaryvalues, that is f = ih with h real-valued. Thus, we have ±G2,0ϕ

±1 = ihϕ±

1 .Since ϕ+

1 and ϕ−1 do not vanish simultaneously, we deduce that h and

G2,0 vanish identically. Thus, C vanishes, too. And the Gauss and Codazziequation are satisfied. Then, we get the conclusion by the fundamentaltheorem of hypersurfaces given above.


Case of signature (1, 1) ωC = −e1e2, hence e1 · e2 · ϕ = −ωC · ϕ = −ϕ. Henceformula (7) becomes

− (R1212 + ε2 det(A) + 4λ2)︸︷︷︸G1,1

ϕ = ε d∇ A(e1, e2)︸︷︷︸C1,1

·ϕ.

or equivalently εC1,1 · ϕ± = G1,1ϕ∓. Applying twice this relation we have

finally

ε2||C1,1||2ϕ± = G21,1ϕ

±.

• Spacelike immersion: ε = 1, M1,1 ↪→ M2,1.

We refer to [5] for the immersion in R2,1. Let us consider the other space forms.

Here again, we need two (RSK)-spinors. Since ϕ±1 do not vanish at the same

point, we have clearly that ||C1,1|| = G21,1 � 0. Moreover, we have

−||C1,1||2 〈ϕ1, ϕ2〉 = ⟨C1,1 · ϕ1, C1,1 · ϕ2

⟩

= −G21,1 〈e1 · e2ϕ1, e1 · e2 · ϕ2〉

= G21,1 〈ϕ1, ϕ2〉 .

Since 〈ϕ1, ϕ2〉 never vanishes, we deduce that ||C1,1|| = −G21,1 � 0. Conse-

quently, ||C1,1|| = G1,1 = 0. Moreover, C1,1 is not isotropic. Indeed, sinceG1,1 = 0, we have C1,1 · ϕ1 = 0 and thus C1,1 automatically vanishes as provedin [5].

• Timelike immersion: ε = i, M1,1 ↪→ M1,2. It is easy to see that computations

similar to the one for the previous case give the result.Two (ISK)-spinors are needed.

Case of Signature (0,2) ωC = −ie1e2, hence e1 · e2 · ϕ = iωC · ϕ = iϕ.Hence formula (7) becomes

i (−R1212 + ε2 det(A) + 4λ2)︸︷︷︸G0,2

ϕ = ε d∇ A(e1, e2)︸︷︷︸C0,2

·ϕ.

or equivalently εC0,2 · ϕ± = ±iG0,2ϕ∓. Applying twice this relation we have

finally

ε2||C0,2||2ϕ± = −G20,2ϕ

±.

• Spacelike immersion ε = 1, M0,2 ↪→ M1,2.

In this case, we have ||C0,2||2ϕ± = −G20,2ϕ

±. Since the metric is negativedefinite, ||C0,2||2 and −G2

0,2 are both non-positive. Hence, we need two(ISK) to use the same argument as for the case M2,0 ↪→ M

2,1.

• Timelike immersion: ε = i, M0,2 ↪→ M0,3.

We get ||C0,2||2ϕ± = G20,2ϕ

±, hence C0,2 = 0 and G20,2 = 0 since the norm

of C0,2 is non-positive. In this case, only one (ISK)-spinor is needed. ��


References

1. Bär, C., Gauduchon, P., Moroianu, A.: Generalized cylinders in semi-Riemannian and spingeometry. Math. Z. 249(3), 545–580 (2005)

2. Baum, H.: Spin-Strukturen und Dirac Operatoren über pseudo-RiemannschenMannigfaltgkeiten. Teubner-Texte zur Mathematik, Bd. 41 Teubner-Verlag, Leipzig (1981)

3. Baum, H., Müller, O.: Codazzi spinors and globally hyperbolic manifolds with specialholonomy. Math. Z. 258(1), 185–211 (2008)

4. Friedrich, T.: On the spinor representation of surfaces in Euclidean 3-space. J. Geom. Phys.28, 143–157 (1998)

5. Lawn, M.A.: Immersions of Lorentzian surfaces in R2,1. J. Geom. Phys. 58(6), 683–700 (2008)

6. Lawn, M.A., Roth, J.: Isometric immersions of Hypersurfaces into 4-dimensional manifoldsvia spinors. Diff. Geom. Appl. 28(2), 205–219 (2010)

7. Lawson, B., Michelson, M.-L.: Spin Geometry. Princeton University Press (1989)8. Morel, B.: Surfaces in S

3 and H3 via spinors. Actes du séminaire de théorie spectrale et

géométrie, vol. 23, pp. 9–22. Institut Fourier, Grenoble (2005)9. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press,

New York (1983)10. Roth, J.: Spinorial characterizations of surfaces into 3-homogeneous manifolds. J. Geom. Phys.

60, 1045–1061 (2010)


Blow-up, Global Existence and Persistence Propertiesfor the Coupled Camassa–Holm equations

Mingxuan Zhu

Received: 7 April 2011 / Accepted: 19 May 2011 / Published online: 8 June 2011© Springer Science+Business Media B.V. 2011

Abstract In this paper, we consider the coupled Camassa–Holm equations.First, we present some new criteria on blow-up. Then global existence andblow-up rate of the solution are also established. Finally, we discuss persistenceproperties of this system.

Keywords Coupled Camassa–Holm equations · Blow-up · Global existence ·Blow-up rate · Persistence properties

Mathematics Subject Classifications (2010) 37L05 · 35Q58 · 26A12

1 Introduction

In this paper, we consider the following model named coupled Camassa–Holmsystem,

⎧⎪⎪⎨

⎪⎪⎩

mt = 2mux + mxu + (mv)x + nvx,

nt = 2nvx + nxv + (nu)x + mux,

u(0, x) = u0(x),

v(0, x) = u0(x),

(1.1)

where m = u − uxx and n = v − vxx.

M. Zhu (B)Department of Mathematics, Zhejiang Normal University, Jinhua 321004,Zhejiang, People’s Republic of Chinae-mail: [email protected]

198 M. Zhu

Let � = (1 − ∂2x)

12 , then the operator �−2 can be expressed by it’s associated

Green’s function G = 12 e−|x| as �−2 f (x) = G ∗ f (x) = 1

2

∫

Re−|x−y| f (y)dy. So

system (1.1) is equivalent to the following system

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ut = (u + v)ux + G ∗ (uvx) + ∂xG ∗(

u2 + 12

u2x + uxvx + 1

2v2 − 1

2v2

x

)

,

vt = (u + v)vx + G ∗ (uxv) + ∂xG ∗(

v2 + 12v2

x + uxvx + 12

u2 − 12

u2x

)

,

u(0, x) = u0(x),

v(0, x) = v0(x).

(1.2)

As far as we known, it seems that system (1.1) appears initially in [10], whichhas the following conserved quantities:

E1(u) =∫

R

udx E2(v) =∫

R

vdx,

E3(u) =∫

R

mdx E4(v) =∫

R

ndx,

E5(u, v) =∫

R

(u2 + v2 + u2

x + v2x

)dx.

It is necessary to point out that system (1.1) has peaked solitons in the formof a superposition of multipeakons. The peakons represent a recent newdevelopment that is of physical interest since peaked traveling waves are thetraveling wave solutions of highest amplitude to the governing equations forwater waves, in contrast to all other traveling waves of smaller amplitude,which are smooth, cf. the discussion in the papers [4, 7, 8, 21].

Local well-posedness theorem for (1.1) was established in [10]. It is provedthat there exists a unique solution (u, v) ∈ C([0, T); Hs × Hs) for any (u0, v0) ∈C([0, T); Hs × Hs) with s > 3

2 .Similar to the standard Camassa–Holm equation, we can show that the

corresponding solution blows up in finite time if and only if

limt→T−

supx∈R

ux(t, x) = ∞ or limt→T−

supx∈R

vx(t, x) = ∞.

In Section 2, we get some new criteria on blow-up, which improved previousresults. A condition for global existence is also fund in Section 2. Blow-uprate is considered in Section 3. In Section 4, persistence properties will beestablished analogous to the standard Camassa–Holm equation.

Remark 1.1 It is worth to point out that recent many works have been donefor similar systems. For details we refer to [11, 12, 16, 19].

Blow-up, Global Existence and Persistence Properties 199

2 Blow-up and Global Existence

First, we establish sufficient conditions on the initial data to guarantee blow-upfor system (1.2).

Theorem 2.1 Assume (u0, v0) ∈ Hs(R) × Hs(R) with s > 32 satisf ies the follow-

ing conditions

u0x(x0) + v0x(x0) �√

32‖(u0, v0)‖H1(R)×H1(R) for some x0 ∈ R.

Then the corresponding solution to (1.1) blows up in f inite time.

Before going to the proof, let us recall the following technique lemma.

Lemma 2.2 Assume that an absolutely continuous function y(t) satisf ies

y′(t) � Cy2(t) − K (2.1)

almost everywhere, with constants C, K > 0. If the initial datum y0 >

√KC , then

the solution to (2.1) goes to ∞ before t tend to 1Cy0− K

y0

.

Because the proof of Lemma 2.2 is easy and analogous to the lemma in [23],so we omit.

Proof Differentiating (1.2) with respect to x, we obtain that⎧⎪⎪⎪⎨

⎪⎪⎪⎩

utx = ∂x((u + v)ux) + ∂xG ∗ (uvx) + ∂2x G ∗

(

u2 + 12

u2x + uxvx + 1

2v2 − 1

2v2

x

)

,

vtx = ∂x((u + v)vx) + ∂xG ∗ (uxv) + ∂2x G ∗

(

v2 + 12v2

x + uxvx + 12

u2 − 12

u2x

)

.

Combining the two equations together, we obtain

∂t(ux+vx) = (ux + vx)2 + (u + v)(uxx + vxx)+∂2

x G ∗(

32

u2+ 32v2+uv+2uxvx

)

.

In view of ∂2x G ∗ F = G ∗ F − F, we have

∂t(ux + vx) = (u + v)2x + (u + v)(uxx + vxx) − F + G ∗ F, (2.2)

where F = 32 u2 + 3

2v2 + uv + 2uxvx. From the following Sobolev embeddinginequality

‖u‖2L∞ � 1

2‖u‖2

H1,

it follows that

F � 2u2 + 2v2 + u2x + v2

x � ‖(u0, v0)‖2H1×H1 .

200 M. Zhu

Additionally, one can get

F ≥ u2 + v2 + 12(u + v)2 + (ux + vx)

2 − u2x − v2

x,

∥∥G ∗ u2

x

∥∥

L∞ � ‖G‖L∞∥∥u2

x

∥∥

L1 = 12

∥∥u2

x

∥∥

L1 ,

∥∥G ∗ v2

x

∥∥

L∞ � ‖G‖L∞∥∥v2

x

∥∥

L1 = 12

∥∥v2

x

∥∥

L1 .

By the above estimates and (2.2), we deduce that

d(ux + vx)

dt� (ux + vx)

2 + (u + v)(uxx + vxx) − 32‖(u0, v0)‖2

H1×H1 .

Applying the idea from [5], one can get that if we let ϕ(t) = supx∈R

(ux + vx)(x, t),

ϕ0 denoting supx∈R

(u0x + v0x)(x), ϕ(t) is absolutely continuous. By the above

analysis, we get

dϕ

dt� ϕ2 − 3

2‖(u0, v0)‖2

H1×H1 .

By applying Lemma 2.2, we have

limt→T

ϕ(t) = +∞ with T = 1

ϕ0 −32 ‖(u0,v0)‖2

H1×H1

ϕ0

.

provided that

ϕ0 >

√32‖(u0, v0)‖H1×H1,

This completes the proof.


ing conditions

u0, v0 is odd and (u0x + v0x)(0) � ‖(u0, v0)‖H1(R)×H1(R).


Proof From (1.1) it is easy to check that if (u(x, t), v(x, t)) is a solution to(1.1) with the initial datum (u0(x), v0(x)), then (−u(−x, t), −v(−x, t)) is also asolution with (−u0(−x), −v0(−x)) being it’s initial datum. Therefore, accordingto the uniqueness of the solution to (1.1),

(u(x, t), v(x, t)) = (−u(−x, t), −v(−x, t)), for all t � 0, x ∈ R.

Taking x = 0 into (2.2) and setting ϕ(t) = (ux + vx)(x = 0, t), we obtain

dϕ

dt= u2

x + v2x + G ∗ F.


It is easy to know

G ∗ F � −12‖(u0, v0)‖2

H1(R)×H1(R),

which implies that

dϕ

dt� 1

2ϕ2 − 1

2‖(u0, v0)‖2

H1(R)×H1(R).

By applying Lemma 2.2 again, we have

limt→T

ϕ(t) = +∞ with T = 2

ϕ0 − ‖(u0,v0)‖2H1×H1

ϕ0

,

provided that

ϕ0 > ‖(u0, v0)‖H1×H1 .

This completes the proof.

Remark 2.1 For G ∗ (u2 + α2

2 u2x), optimal constant for the following inequality

was established in [17, 24]:

G ∗(

u2 + α2

2u2

x

)

� Cαu2(x),

with

Cα = 12

+ arctan(sinh( 12α

))

2 sinh( 1

2α

) + 2 arctan(sinh

( 12α

))sinh2 ( 1

2α

) .

But here our problem is different from theirs.Motivated by the idea from [1, 9, 10, 22], we also have the following theorem

via the integral form of initial value:


ing conditions∫

R

(u0x + v0x)3dx � 2

√3‖(u0, v0)‖H1×H1 .


Proof Multiplying equation (2.2) by (ux + vx)2 and integrating with respect to

x, we get

13

ddt

∫

R

(ux + vx)3dx =

∫

R

(ux + vx)4dx +

∫

R

(u + v)(uxx + vxx)(ux + vx)2dx

−∫

R

(ux + vx)2(F − G ∗ F)dx. (2.3)

202 M. Zhu

For the last term, we know∫

R

(ux + vx)2(F − G ∗ F)dx � 2‖(u, v)‖2

H1×H1

(‖F‖L∞ + ‖G ∗ F‖L∞)

� 2‖(u, v)‖2H1×H1

(2‖(u, v)‖2

H1×H1

)

� 4‖(u, v)‖2H1×H1,

here using the estimates that

‖ f‖L∞ � 12‖ f‖H1 and |G ∗ f | � 1

2‖ f‖L1 .

According to Cauchy–Schwartz inequality,∣∣∣∣

∫

R

(ux + vx)3dx

∣∣∣∣ �

(∫

R

(ux + vx)2dx

) 12(∫

R

(ux + vx)4dx

) 12

,

it follows that(∫

R

(ux + vx)4dx

)

� 12

(‖(u, v)‖2H1×H1

)

(∫

R

(ux + vx)3dx

)2

.

Therefore, putting it into (2.3), we get

ddt

∫

R

(ux + vx)3dx �

∫

R(ux + vx)

3dx

‖(u0, v0)‖2H1×H1

− 12‖(u0, v0)‖4H1×H1 . (2.4)

Let ϕ(t) = ∫

R(ux + vx)

3dx, ϕ(t) is absolutely continuous. we can rewrite(2.4) as

dϕ

dt� ϕ2

‖(u0, v0)‖2H1×H1

− 12‖(u0, v0)‖4H1×H1 .

In view of Lemma 2.2, we have

limt→T

ϕ(t) = ∞ with T = 1

‖(u0, v0)‖2H1×H1ϕ0 − ‖(u0,v0)‖4

H1×H1

ϕ0

,

provided

ϕ(0) > 2√

3‖(u0, v0)‖3H1×H1,

On the other hand, since∫

R

(ux + vx)3dx � ‖ux + vx‖L∞

∫

R

(ux + vx)2dx

� 2‖ux + vx‖L∞‖(u0, v0)‖2H1×H1 .

show that

limt→T

∫

R

(ux + vx)3dx = ∞ and lim

t→Tsupx∈R

‖ux + vx‖L∞ = ∞.

This complete the proof.


Remark 2.2 In [10], the condition is(∫

R

(u0x + v0x)3dx

)2

� 48(4c + 3)‖(u0, v0)‖2H1×H1,

where c is a constant which satisfies ‖ f‖L∞ � c‖ f‖H1 . To the non-periodiccase, c = 1

2 . So Theorem 2.4 is an improvement of that in [10].

Inspired by the considerations about the Camassa–Holm equation made inthe papers [2, 6]. The following theorem will show us that system (1.2) alsoadmits global existence.

Theorem 2.5 Suppose that (u0, v0) ∈ Hs(R) × Hs(R) with s > 32 , m0 + n0

doesn’t change sign. Then the corresponding solution to (1.1) exists globally.

Proof If we assume that m0 + n0 > 0, it is sufficient to prove ux(x, t) + vx(x, t)has a supper bound for all t. In fact

ux(x, t) = −12

e−x∫ x

−∞eξ m(ξ, t)dξ + 1

2e−x

∫ ∞

xeξ m(ξ, t)dξ,

vx(x, t) = −12

e−x∫ x

−∞eξ n(ξ, t)dξ + 1

2e−x

∫ ∞

xeξ n(ξ, t)dξ,

Combining the two equations together, we get

ux(x, t) + vx(x, t) � 12

e−x∫ ∞

xe−ξ m(ξ, t)dξ + 1

2e−x

∫ ∞

xeξ n(ξ, t)dξ

� 12

∫ ∞

xm(ξ, t) + n(ξ, t)dξ

� 12

∫ ∞

−∞m(ξ, t) + n(ξ, t)dξ

= 12

∫ ∞

−∞m0(ξ, t) + n0(ξ, t)dξ.

This complete the proof.

3 Blow up Rate

If we know the solution blows up in finite time, it is natured to consider theblow-up profile. Usually, it is very difficult, but the blow-up rate [6] withrespect to time for (1.2) can be show as following.

Theorem 3.1 Assume that (u0, v0) ∈ Hs(R) × Hs(R) with s > 32 , (u, v) is the

corresponding solution. If the lifespan of the solution is f inite, then

limt→T

(T − t) supx∈R

(ux + vx)(x, t) = 1.

204 M. Zhu

Proof Let ϕ(t) = supx∈R(ux + vx)(x, t), then we can rewrite (2.2) as

dϕ

dt− ϕ2 = −F + G ∗ F.

Considering the following inequality

| − F + G ∗ F| � |F| + |G ∗ F| � 2‖(u0, v0)‖2H1×H1 = K,

one can get

−K � dϕ

dt− ϕ2 � K,

where K is a constant depending on ‖(u0, v0)‖2H1×H1 .

Since limt→T

ϕ(t) = ∞, it follows that for any ε ∈ (0, 1), there exists a t0 such

that ϕ2(t) > Kε

for all t ∈ (t0, T). Therefore,

ϕ2 − εϕ2 � dϕ

dt� ϕ2 + εϕ2,

which means that

1 − ε � 1ϕ2

dϕ

dt� 1 + ε.

Directly integrating from t to T gives us

11 + ε

� (T − t)ϕ(t) � 11 − ε

.

Because of the arbitrariness of ε, this completes our proof.

4 Persistence Properties

Motivated by Mckean’s deep observation for the Camassa–Holm equation[18], we can do the similar particle trajectory as

{qt = −(u(q, t) + v(q, t)), 0 < t < T, x ∈ R,q(x, 0) = x, x ∈ R,

(4.1)

where T is the life span of the solution, then q is a diffeomorphism of the line.Differentiating the first equation in (4.1) with respect to x, one has

dqt

dx= qxt = −(ux(q, t) + vx(q, t))qx, t ∈ (0, T).

Hence

qx(x, t) = exp{∫ t

0−(ux(q, s) + vx(q, s))ds

}

, qx(x, 0) = 1.

Now, we have the following lemma that the potential m − n with compactlysupported initial datum m0 − n0 also has compact x−support as long as it ex-ists. Similar results to the Camassa–Holm equation can be found in [3, 13, 14].More precisely,


Lemma 4.1 Assume that (u0, v0) ∈ Hs(R) × Hs(R) with s > 32 , (u, v) is the

corresponding solution, if m0 − n0 has compact support, then m − n also hascompact support.

Proof Since

ddt

(m(q)q2

x

) = [mt(q) − mx(q)(u(q, t) + v(q, t)) − m(q)(ux(q, t) + vx(q, t))]q2x

= mux + nvx.

Similarly,

ddt

(n(q)q2

x

) = mux + nvx,

so it follows that

(m(q) − n(q))q2x = m0(x) − n0(x).

Now, we shall investigate the following property for the strong solutionsto (1.2) in L∞−space which asymptotically exponential decay at infinity astheir initial profiles. The main idea comes from a recent work of Zhou and hiscollaborators [15] for the standard Camassa–Holm equation (for slower decayrate, we refer to [20] ).

Theorem 4.2 Assume that for some T > 0 and s > 52 , (u, v) ∈ C([0, T]; Hs(R)×

Hs(R)) is a strong solution of (1.2) and that (u0(x), v0(x)) = (u(x, 0), v(x, 0))

satisf ies that for some θ ∈ (0, 1),

|u0(x)|, |u0x(x)|, |v0(x)|, |v0x(x)| ∼ O(e−θx) .

Then

|u(x, t)|, |ux(x, t)|, |v(x, t)|, |vx(x, t)| ∼ O(e−θx)

uniformly in the time interval [0, T].

Proof First, we shall introduce the weight function to get the desired result.This function ϕN(x) with N ∈ Z

+ is independent on t as follows:

ϕN(x) =⎧⎨

⎩

1, x � 0,eθx, x ∈ (0, N),eθ N, x � N,

which implies that

0 � ϕ′N(x) � ϕN(x).

From the first equation in system (1.2), we can get

∂t(uϕN) = (uϕN)ux + ϕN S, (4.2)

206 M. Zhu

where S = vux + G ∗ (uvx) + ∂xG ∗ (u2 + 12 u2

x + uxvx + 12v2 − 1

2v2x). Multiply-

ing (4.2) by (uϕN)2p−1 with p ∈ Z+ and integrating the result in the x−variable,

we get∫ +∞

−∞∂t(uϕN)(uϕN)2p−1dx=

∫ +∞

−∞(uϕN)ux(uϕN)2p−1dx+

∫ +∞

−∞ϕN S(uϕN)2p−1dx,

from which we can deduce thatddt

‖uϕN‖L2p � ‖ux‖L∞‖uϕN‖L2p + ‖ϕN S‖L2p .

Denoting M = supt∈[0,T]

‖(u(t), v(t))‖Hs and by the Gronwall’s inequality, we ob-

tain

‖uϕN‖L2p �(

‖u0ϕN‖L2p +∫ t

0‖ϕN S‖L2p dτ

)

eMt. (4.3)

Taking the limits in (4.3), we get

‖uϕN‖L∞ �(

‖u0ϕN‖L∞ +∫ t

0‖ϕN S‖L∞dτ

)

eMt. (4.4)

Similarly, we can get

‖vϕN‖L∞ �(

‖v0ϕN‖L∞ +∫ t

0

∥∥ϕN S

∥∥

L∞dτ

)

eMt. (4.5)

where S = uvx + G ∗ (uxv) + ∂xG ∗ (v2 + 12v2

x + uxvx + 12 u2 − 1

2 u2x). Next

differentiating the first equation in (1.2) in the x−variable produces theequation

utx = uuxx + u2x + ∂xS. (4.6)

Using the weight function, we can rewrite (4.6) as

∂x(uxϕN) = uuxxϕN + (uxϕN)ux + ϕN∂xS. (4.7)

Multiplying (4.7) by (uxϕN)2p−1 with p ∈ Z+ and integrating the result in the

x−variable, it follows that∫ +∞

−∞∂t(uxϕN)(uxϕN)2p−1dx =

∫ +∞

−∞uuxxϕN(uxϕN)2p−1dx

+∫ +∞

−∞(uxϕN)ux(uxϕN)2p−1dx +

∫ +∞

−∞ϕN∂xS(uxϕN)2p−1dx. (4.8)

For the first term on the right side of (4.8), we know∣∣∣∣

∫ +∞

−∞uuxxϕN(uxϕN)2p−1dx

∣∣∣∣ � 2

(‖u‖L∞ + ‖ux‖L∞)‖uxϕN‖2p

L2p .

Using the above estimate and Hölder inequality, we deduce that

ddt

‖uxϕN‖L2p � 5M‖uxϕN‖L2p + ‖ϕN∂xS‖L2p .


Thanks to the Gronwall’s inequality, it holds that

‖uxϕN‖L2p �(‖u0xϕN‖L2p + ‖ϕN∂xS‖L2p

)e5Mt. (4.9)

Taking the limits in (4.9), we have

‖uxϕN‖L∞ �(

‖u0xϕN‖L∞ +∫ t

0‖ϕN∂xS‖L∞dτ

)

e5Mt. (4.10)


‖vxϕN‖L∞ �(

‖v0xϕN‖L∞ +∫ t

0

∥∥ϕN∂x S

∥∥

L∞dτ

)

e5Mt. (4.11)

Combining (4.4), (4.5), (4.10) and (4.11) together, it follows that

‖uϕN‖L∞ + ‖vϕN‖L∞ + ‖uxϕN‖L∞ + ‖vxϕN‖L∞

�(‖u0ϕN‖L∞ + ‖v0ϕN‖L∞ + ‖u0xϕN‖L∞ + ‖v0xϕN‖L∞

)e5Mt

+ e5Mt(∫ t

0‖ϕN S‖L∞ + ∥

∥ϕN S∥∥

L∞ + ‖ϕN∂xS‖L∞ + ∥∥ϕN∂x S

∥∥

L∞dτ

)

.

(4.12)

A simple calculation shows that there exists c0 > 0, depending only on θ ∈(0, 1), such that for any N ∈ Z

+,

ϕN(x)

∫ ∞

−∞e−|x−y| 1

ϕN(y)dy � c0 = 4

1 − θ.

Thus, for any appropriate function f and g one sees that

|ϕNG ∗ f (x)g(x)| =∣∣∣∣12ϕN(x)

∫ ∞

−∞e−|x−y| f (y)g(y)dy

∣∣∣∣

� 12ϕN(x)

∫ ∞

−∞e−|x−y| 1

ϕN(y)ϕN(y) f (y)g(y)dy

� 12

(

ϕN(x)

∫ ∞

−∞e−|x−y| 1

ϕN(y)dy

)

‖ϕN f‖L∞‖g‖L∞

� c0‖ϕN f‖L∞‖g‖L∞ .


|ϕN∂xG ∗ f (x)g(x)| � c0‖ϕN f‖L∞‖g‖L∞ ,

and

|ϕN∂2x G ∗ f (x)g(x)| � c0‖ϕN f‖L∞‖g‖L∞ .

208 M. Zhu

Thus, inserting the above estimates into (4.12), there exists a constant c =c(M, T, c) � 0 such that


� c(‖u0ϕN‖L∞ + ‖v0ϕN‖L∞ + ‖u0xϕN‖L∞ + ‖v0xϕN‖L∞

)

+ c∫ t

0(‖u‖L∞ + ‖v‖L∞ + ‖ux‖L∞ + ‖vx‖L∞ + ‖uxx‖L∞ + ‖vxx‖L∞)

(‖uϕN‖L∞ + ‖vϕN‖L∞ + ‖uxϕN‖L∞ + ‖vxϕN‖L∞dτ


+ c∫ t

0

(‖uϕN‖L∞ + ‖vϕN‖L∞ + ‖uxϕN‖L∞ + ‖vxϕN‖L∞dτ.

Hence, for any t ∈ Z+ and any t ∈ [0, T], we have



)

� c(∥∥u0 max

(1, eθx)∥∥

L∞ + ∥∥v0 max

(1, eθx)∥∥

L∞

+ ∥∥u0x max

(1, eθx)∥∥

L∞ + ∥∥v0x max

(1, eθx)∥∥

L∞ . (4.13)

Finally, taking the limit as N goes to infinity in (4.13) we find that for anyt ∈ [0, T]

∥∥ueθx

∥∥

L∞ + ∥∥veθx

∥∥

L∞ + ∥∥uxeθx

∥∥

L∞ + ∥∥vxeθx

∥∥

L∞

� c(∥∥u0 max

(1, eθx)∥∥

L∞ + ∥∥v0 max

(1, eθx)∥∥

L∞

+ ∥∥u0x max

(1, eθx)∥∥

L∞ + ∥∥v0x max

(1, eθx)∥∥

L∞ ,

which completes the proof of the theorem. Acknowledgements The author would like to thank the referee for valuable comments andsuggestions. This work is partially supported by Zhejiang Innovation Project (T200905), ZJNSF(Grant No. R6090109) and NSFC (Grant No. 10971197).

References

1. Constantin, A.: On the Cauchy problem for the periodic Camassa–Holm equation. J. Differ.Equ. 141, 218–235 (1997)

2. Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: ageometric approach. Ann. Inst. Fourier 50, 321–362 (2000)

3. Constantin, A.: Finite propagation speed for the CamassaHolm equation. J. Math. Phys. 46,023506 (2005)

4. Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math 166, 523–535 (2006)5. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equation. Acta

Math. 181, 229–243 (1998)6. Constantin, A., Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a

shallow water equation. Math. Z. 233, 75–91 (2000)


7. Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc.44, 423–431 (2007)

8. Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves withvorticity. Math. Ann. 173, 559–568 (2011)

9. Fu, Y., Liu, Y., Qu, C.: Well-possdness and blow-up solution for a modified two-componentCamassa–Holm system with peakons. Math. Ann. 348, 415–448 (2010)

10. Fu, Y., Qu, C.: Well-possdness and blow-up solution for a new coupled Camassa–Holmequations with peakons. J. Math. Phys. 50, 012906 (2009)

11. Guo, Z.: Blow up and global solutions to a new integrable model with two components.J. Math. Anal. Appl. 372, 316–327 (2010)

12. Guo, Z., Zhou, Y.: On solutions to a two-component generalized Camassa–Holm equation.Stud. Appl. Math. 124, 307–322 (2009)

13. Henry, D.: Compactly supported solutions of the Camassa–Holm equation. J. Nonlin. Math.Phys. 12, 342–347 (2005)

14. Henry, D.: Persistence properties for a family of nonlinear partial differential equations.Nonlinear Anal. 70, 1565–1573 (2009)

15. Himonas, A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continua-tion of solutions of the Camassa–Holm equation. Commun. Math. Phys. 271, 511–522 (2007)

16. Jin, L., Guo, Z.: On a two-component Degasperis-Procesi shallow water system. NonlinearAnal. 11, 4164–4173 (2010)

17. Jin, L., Liu, Y., Zhou, Y.: Blow-up of solutions to a periodic nonlinear dispersive rod equation.Doc. Math. 15, 267–283 (2010)

18. Mckean, H.P.: Breakdown of a shallow water equation. Asian J. Math. 2, 767–774 (1998)19. Ni, L.: The Cauchy problem for a two-component generalized θ -equations. Nonlinear Anal.

73, 1338–1349 (2010)20. Ni, L., Zhou, Y.: A new asymptotic behavior of solutions to the Camassa–Holm equation.

Proc. Am. Math. Soc. (2011). doi:10.1007/s11040-011-9094-221. Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)22. Zhou, Y.: Wave breaking for a shallow water equation. Nonlinear Anal. 57, 137–152 (2004)23. Zhou, Y.: Blow-up phenomenon for a periodic rod equation. Phys. Lett. A 353, 479–486 (2006)24. Zhou, Y.: Blow-up of solutions to the DGH equation. J. Funct. Anal. 250(1), 227–248 (2007)

http://dx.doi.org/10.1007/s11040-011-9094-2


Formulas and Asymptotics for the Asymmetric SimpleExclusion Process

Craig A. Tracy · Harold Widom

Received: 25 February 2011 / Accepted: 22 May 2011 / Published online: 11 June 2011© Springer Science+Business Media B.V. 2011

Abstract This is an expanded version of a series of lectures delivered by thesecond author in June, 2009. It describes the results of three of the authors’papers on the asymmetric simple exclusion process, from the derivation ofexact formulas for configuration probabilities, through Fredholm determinantrepresentation, to asymptotics with step initial condition establishing KPZuniversality. Although complete proofs are in general not given, at least themain elements of them are.

Keywords Asymmetric simple exclusion process · Fredholm determinant ·Asymptotics · KPZ universality

Mathematics Subject Classification (2010) 60K35

1 Introduction

The asymmetric simple exclusion process (ASEP) is a special case of processesintroduced in 1970 by F. Spitzer [8]. In ASEP, particles are at integer sites onthe line. Each particle waits exponential time, and then

This is an expanded version of a series of lectures delivered by the second author atUniversité de Paris in June, 2009, describing the results in the articles [9–11]. Althoughcomplete proofs will in general not be presented here, at least the main elements of themwill be.

C. A. TracyDepartment of Mathematics, University of California, Davis, CA 95616, USA

H. Widom (B)Department of Mathematics, University of California, Santa Cruz, CA 95064, USAe-mail: [email protected]

212 C.A. Tracy, H. Widom

(1) with probability p it moves one step to the right if the site is unoccupied,otherwise it does not move;

(2) with probability q = 1 − p it moves one step to the left if the site isunoccupied, otherwise it does not move.

In the totally asymmetric simple exclusion process (TASEP) particles can onlymove in one direction, so either p = 0 or q = 0. In a major breakthrough,K. Johansson [2] related a probability in TASEP to a probability in randommatrix theory. If q = 0 and initially particles were at the negative integers,then the probability that at time t the particle initially at −m has moved at leastn (� m) times equals the probability distribution for the largest eigenvalue inthe Laguerre ensemble of m × m matrices with weight function xn−m e−x. Thus,it is given by a constant depending on m and n times the determinant

det(∫ t

0xn−m+i+ j e−x dx

)i, j=0,...,m−1

.

This connection led to considerable progress in understanding TASEP and thederivation of asymptotic results. For ASEP there is no longer a determinantalstructure and a different approach was required.

Here are the main results of [9–11]. First we consider ASEP with finitelymany particles. For N-particle ASEP a possible configuraion is given by

X = {x1, . . . , xN}, x1 < · · · < xN, (xi ∈ Z).

Thus the xi are the occupied sites. We assume an initial configuration Y ={y1, . . . , yN}, and obtain formulas for

(1) PY(X; t), the probability that at time t the system is in configuration X.1

(2) PY(xm(t) = x), the probability that at time t the mth particle from the leftis at x.

The formula we get for the latter extends to infinite systems

y1 < y2 < · · · → +∞.

In particular we may take Y = Z+. (This is the step initial condition.)

For the derivation of (1) we use the Bethe Ansatz [1] to obtain a solution of adifferential equation with boundary conditions. The derivation of (2) from (1)requires the proof of two combinatorial identities. The derivation we outlinefor this is from [12] and simpler than the one in [9].

For step initial condition we show that P(xm(t) = x) has a representation interms of Fredholm determinants. This makes asymptotic analysis possible. Weassume that q > p, so there is a drift to the left, define γ = q − p, and obtainasymptotics as t → ∞ for

P (xm(t/γ ) > x) (fixed m and x),

1This had been known for the case N = 2 [7].

Formulas and Asymptotics for Simple Exclusion Process 213

and the limits as t → ∞ of

P(xm(t/γ ) � −t + γ 1/2 s t1/2) (m fixed),

P(xm(t/γ ) � c1(σ ) t + c2(σ ) s t1/3) (m = [σ t]),

where c1(σ ) and c2(σ ) are certain explicit constants.The last limit is the distribution function F2(s) of random matrix theory, the

limiting distribution for the rescaled largest eigenvale in the Gaussian unitaryensemble. These asymptotics were obtained in [2] for the case of TASEP.(That F2 should arise in ASEP had long been suspected. This is referred toas KPZ universality [3].)

2 Integral Formulas

2.1 The Differential Equation

The idea goes back to [1]. There is a differential equation with boundaryconditions and an initial condition whose solution gives PY(X; t). To state itwe introduce the new notation u(X; t) or u(X) in place of PY(X; t).2

We first consider the case N = 2, and consider du(x1, x2)/dt. After anexponential waiting time, the system could enter state {x1, x2} or it couldleave this state. Assume first that x2 − x1 > 1, so that there is no interferencebetween the two particles. The system could enter the state if the first particlehad been at x1 − 1 (this has probability u(x1 − 1, x2)) and moved one step tothe right (probability p), and three other analogous ways. The system couldleave the state if the first particle is at x1 (probability u(x1, x2)) and movesone step to the right (probability p) or one step to the left (probability q), andanalogously for the second particle. These give the equation

ddt

u(x1, x2) = p u(x1 − 1, x2) + q u(x1 + 1, x2)

+ p u(x1, x2 − 1) + q u(x1, x2 + 1) − 2 u(x1, x2).

But if x2 − x1 = 1 then for entering the state the first particle could not havebeen one step to the right nor the second particle one step to the left, and forleaving the state the first particle cannot move right nor can the second particlemove left. Therefore in this case

ddt

u(x1, x2) = p u(x1 − 1, x2) + q u(x1, x2 + 1) − u(x1, x2).

2The reason is that if X = {x1, . . . , xN} then PY (X; t) only makes sense when x1 < · · · < xN , butfor u(X; t) there will be no such requirement.


We could combine these two equations into one, but then the right sidewould have nonconstant coefficients. Instead, as in [1], we observe that if weformally subtract the two equations we get, when x2 = x1 + 1,

0 = p u(x1, x1) + q u(x1 + 1, x1 + 1) − u(x1, x1 + 1).

If the first equation holds for all x1 and x2, and this last boundary conditionholds for all x1, then the second equation holds when x2 = x1 + 1. So anequation with nonconstant coefficients has been replaced with an equationwith constant coefficients plus a boundary condition.

This was done for N = 2, but it holds for general N. Suppose the functionu(X; t), defined for all X = {x1, . . . , xN} ∈ Z

N , satisfies the master equation

ddt

u(X; t) =N∑

i=1

[p u(. . . , xi − 1, . . .) + q u(. . . , xi + 1, . . .) − u(X)],

and the boundary conditions

u(. . . , xi, xi + 1, . . .) = p u(. . . , xi, xi, . . .) + q u(. . . , xi + 1, xi + 1, . . .),

for i = 1, . . . , N − 1.3 Suppose also that it satisfies the initial condition

u(X; 0) = δY(X) when x1 < · · · < xN,

which reflects the initial configuration Y. Then

u(X; t) = PY(X; t) when x1 < · · · < xN.

Thus the strategy will be: (1) find a large class of solutions to the masterequation; (2) find a subset satisfying the boundary conditions; (3) find one ofthese satisfying the initial condition. The last will be the hard (and new) part.

2.2 Solutions to the Master Equation

Define

ε(ξ) = p ξ−1 + q ξ − 1.

For any nonzero complex numbers ξ1, . . . , ξN , a solution of the equation is∏i

(ξ

xii eε(ξi) t) .

Since the ξi are arbitrary another solution is obtained by permuting them.Thus, for any σ in the symmetric group SN another solution is∏

i

ξxiσ(i)

∏i

eε(ξi) t.

3For N ≥ 3 the boundary conditions arising from configurations with more than two adjacentparticles automatically follow from the boundary conditions arising from two adjacent particles.


(The second factor is symmetric in the ξi, which is why we can write it as wedo.) Since the equation is linear, any linear combination of these is a solution,as is any integral (over the ξi) of a linear combination. Thus we arrive at theBethe Ansatz solutions

u(X; t) =∫ ∑

σ∈SN

Fσ (ξ)∏

i

ξxiσ(i)

∏i

eε(ξi) t dNξ. (1)

The Fσ are arbitrary functions of the ξi, and the domain of integration isarbitrary.

2.3 Satisfying the Boundary Conditions

We look for functions Fσ such that the integrand satisfies the boundaryconditions pointwise. The ith boundary condition is satisfied pointwise when

∑σ∈SN

Fσ (p + q ξσ(i) ξσ(i+1) − ξσ(i+1)) (ξσ(i)ξσ(i+1))xi

∏j�=i, i+1

ξx j

σ( j) = 0. (2)

Define Tiσ to be the permutation that differs from σ by an intechange ofthe ith and (i + 1)st entries. Thus, if σ = (2 3 1 4) then T2σ = (2 1 3 4). Since Ti

is bijective, (2) is unchanged if each σ in the summand is replaced by Tiσ ,and therefore unchanged if we add the two. Since the last two factors areunchanged upon replacing σ by Tiσ , we see that a sufficient condition that(2) satisfied is that for each σ ,

Fσ (p + qξσ(i)ξσ(i+1) − ξσ(i+1)) + FTiσ (p + qξσ(i)ξσ(i+1) − ξσ(i)) = 0.

Because the expression will appear so often it is convenient to define

f (ξ, ξ ′) = p + qξξ ′ − ξ,

so the condition becomes

Fσ f (ξσ(i+1), ξσ(i)) + FTiσ f (ξσ(i), ξσ(i+1)) = 0.

This is to hold for all σ and all i. Since these are (n − 1) n! linear equationsin the n! unknowns Fσ , we cannot necessarily expect a solution. But there aresolutions, and in fact it is easy to see that

Fσ (ξ) = sgn σ∏i< j

f (ξσ(i), ξσ( j)) × ϕ(ξ),

where ϕ is an arbitrary function of the ξi, satisfies the equations. (In fact this isthe general solution, since if any Fσ is known then all others are determined.)These Fσ in (1) give a family of solutions to the master equation that satisfythe boundary conditions.


2.4 Satisfying the Initial Condition

The initial condition is∫ ∑σ∈SN

Fσ (ξ)∏

i

ξxiσ(i) dNξ = δY(X). (3)

We begin with the fact that if C is a contour enclosing zero then4

∫C

ξ x−y−1 dξ = δy(x).

Therefore ∫CN

∏i

ξxi−yi−1i dNξ = δY(X).

Thus if id denotes the identity permutation, then the σ = id summand in (3)will give the integral δY(X) if the integration is over CN and

Fid(ξ) =∏

i

ξ−yi−1i .

For this to hold we choose

ϕ(ξ) =∏i< j

f (ξi, ξ j)−1 ·

∏i

ξ−yi−1i .

If we define

Aσ = sgn σ

∏i< j f (ξσ(i), ξσ( j))∏

i< j f (ξi, ξ j)(4)

then the solution we have chosen is

u(X; t) =∑

σ

∫CN

Aσ (ξ)∏

i

ξxiσ(i)

∏i

(ξ

−yi−1i eε(ξi) t) dNξ. (5)

It satisfies the master equation and boundary conditions, and the σ = idsummand satisfies initial condition.

Observe that because of the poles of Aσ when σ �= id this will not be well-defined until we specify C further.

2.4.1 TASEP

When p = 1 we have

Aσ = sgn σ∏

i

(1 − ξσ(i))σ(i)−i.

4Unless specified otherwise, all contours are described once counterclockwise, and all contourintegrals have a factor 1/2π i.


Because of this product structure the integrals of products in (5) may be writtenas product of integrals and (5) becomes

u(X; t) =∑

σ

sgn σ∏

i

∫C(1 − ξσ(i)−i) ξ xi−yσ(i)−1 e(ξ−1−1)t dξ

= det(∫

C(1 − ξ) j−i ξ xi−y j−1e(ξ−1−1)t dξ

).

Schütz [7] obtained this solution to the master equation, using Bethe Ansatzas we have described, and went further to show that it satisfies the initialcondition when the point ξ = 1 is outside the contour C. So he established theformula

PY(X; t) = det(∫

Cr

(1 − ξ) j−i ξ xi−y j−1e(ξ−1−1)t dξ

),

where Cr denotes the circle with center zero and radius r < 1.

2.4.2 ASEP

In [7] Schütz also considered ASEP and showed that when N = 2 theprobability PY(X; t) is equal to a sum of a two-dimensional integral and aone-dimensional integral. In the two-dimensional integral the contours weredifferent. It turns out that if one integrates over small contours only thenthe sum is the sum of two two-dimensional integrals. And this extends togeneral N.

Recall that because of the poles of Aσ , it matters which contours C we takein (5). When p �= 0 all poles of the Aσ will lie outside Cr if r is small enough.These are the contours we take.

Theorem [9, Theorem 2.1] Suppose p �= 0 and assume that r is so small that allpoles of the Aσ lie outside Cr. Then

PY(X; t) =∑

σ

∫CN

r

Aσ (ξ)∏

i

ξxiσ(i)

∏i

(ξ

−yi−1i eε(ξi) t) dNξ. (6)

For the proof we have to show that the initial condition is satisfied. Sincethe σ = id summand satisfies the initial condition, what is to be shown isthat if

I(σ ) =∫CN

r

Aσ (ξ)∏

i

ξxiσ(i)

∏i

ξ−yi−1i dNξ,


then ∑σ �=id

I(σ ) = 0.5 (7)

We show that the permutations in SN\{id} can be grouped in such a way thatthe sum of the I(σ ) from each group is equal to zero. For 1 � n < N fix n − 1distinct numbers i1, . . . , in−1 ∈ [1, N − 1], define A = {i1, . . . , in−1}, and then

SN(A) = {σ ∈ SN : σ(1) = i1, . . . , σ (n − 1) = in−1, σ (n) = N}.When n = 1 these are all permutations with σ(1) = N. When n = N − 1 eachSN(A) consists of a single permutation. Let B be the complement of A ∪ {N}in [1, N].

We first we make the substitution

ξN → η∏i<N ξi

,

in all the integrals. The product of the powers of the ξi in (6) becomes

ηxn−yN−1∏i<N

ξx

σ−1(i)−xn+yN−yi−1i . (8)

We use the alternative representation

Aσ =∏<k

σ−1()>σ−1(k)

(− f (ξk, ξ)

f (ξ, ξk)

)(9)

to see what happens when we shrink some of the ξi-contours. The only poleswe might cross come from the denominatiors in (9) after the substitution, andthese are in the ξi-variables when i ∈ B.

Lemma 1 When n = N − 1 we have I(σ ) = 0 for σ ∈ SN(A).

There is a single i ∈ B and in this case there is no pole in the ξi-variablecoming from (9). Using xN > xN−1 and yN > yi in (8), we see that the integrandis analytic at ξi = 0. Therefore the integral with respect to ξi is zero.

Lemma 2 When n < N − 1 all I(σ ) with σ ∈ SN(A) are sums of lower-orderintegrals in each of which a partial product in (9) independent of σ ∈ A isreplaced by another factor. In each integral some ξi with i ∈ B is equal to anotherξ j with j ∈ B.

5There was an error in the proof of this in [9]. The correction to the proof is to be found in [13].We outline the proof here.


If j = max B, we shrink the ξ j-contour and obtain (N − 1)-dimensionalintegrals coming from poles associated with the variables ξk with k ∈ B\{ j}.For each such k we integrate with respect to ξk the residue at this pole byshrinking the contour, and obtain (N − 2)-dimensional integrals having theproperty described in the lemma.

Lemma 3 For each integral of Lemma 2 there is a partition of SN(A) into pairsσ, σ ′ such that I(σ ) + I(σ ′) = 0 for each pair.

Consider an integral in which ξi = ξ j. We pair σ and σ ′ if σ−1(i) = σ ′−1( j)

and σ−1( j) = σ ′−1(i), and σ−1(k) = σ ′−1

(k) when k �= i, j. The factor (8) isclearly the same for both when ξi = ξ j, and Aσ and Aσ ′ are negatives of eachother then.

Here is why. Assume for definiteness that i < j and σ−1(i) < σ−1( j). Thenthe factor corresponding to = i and k = j does not appear for σ in (9) but itdoes appear for σ ′. This factor equals −1 when ξi = ξ j. And it is straightforwardto check that for any k �= i, j the product of factors involving k and either i or jis the same for σ and σ ′ when ξi = ξ j.

Now (7) can be shown by induction on N. When N = 2 it follows fromLemma 1. Assume N > 2 and that the result holds for N − 1. For thosepermutations for which σ(N) = N we integrate with respect to ξ1, . . . , ξN−1and use the induction hypothesis. The set of permutations with σ(N) < N isthe disjoint union of the various SN(A), and for these we apply Lemmas 1and 3.

2.5 The Left-most Particle

The probability PY(x1(t) = x) is the sum of P(X; t) over all X for which x1 =x, thus over all x2, . . . , xN statisfying x < x2 < · · · < xN < ∞. When r < 1 wemay sum under the integral sign in (6), and the integrand becomes

∏i(ξ

x−yi−1i eε(ξi)t)∏

i< j f (ξi, ξ j)

·∑

σ

sgn σ

⎛⎝∏

i< j

f (ξσ(i), ξσ( j))

× ξσ(2)ξ2σ(3) · · · ξ N−1

σ(N)

(1 − ξσ(2)ξσ(3) · · · ξσ(N))(1 − ξσ(3) · · · ξσ(N)) · · · (1 − ξσ(N))

).


Fortunately we have our first combinatorial identity,6∑σ

sgn σ

×⎛⎝∏

i< j

f (ξσ(i), ξσ( j)) · ξσ(2)ξ2σ(3) · · · ξ N−1

σ(N)

(1−ξσ(1)ξσ(2) · · · ξσ(N))(1−ξσ(2) · · · ξσ(N)) · · · (1−ξσ(N))

⎞⎠

= pN(N−1)/2

∏i< j(ξ j − ξi)∏

i(1 − ξi). (10)

Therefore we obtain

Theorem [9, Theorem 3.1] If p �= 0 and r is so small that all poles of the Aσ lieoutside Cr, then

PY(x1(t) = x) = pN(N−1)/2∫CN

r

∏i< j

ξ j − ξi

f (ξi, ξ j)

1 − ξ1 · · · ξN∏i(1 − ξi)

∏i

(ξx−yi−1i eε(ξi)t) dNξ.

Identity (10) is proved by induction on N. Call the left side of the identityϕN(ξ1, . . . , ξN) and the right side ψN(ξ1, . . . , ξN), and assume the identity holdsfor N − 1. We first sum over all permutations such that σ(1) = k, and then sumover k. If we observe that the inequality i < j becomes j �= i when i = 1, we seethat what we get for the left side of (10), using the induction hypothesis, is

11 − ξ1 ξ2 · · · ξN

N∑k=1

(−1)k+1∏j�=k

f (ξk, ξ j)

·∏j�=k

ξ j · ψN−1(ξ1, . . . , ξk−1, ξk+1, . . . , ξN).

If we substitute for ψN−1(ξ1, . . . , ξk−1, ξk+1, . . . , ξN) what it is and do somealgebra, we find this would equal the right side of (10) if a simpler identityheld:

N∑k=1

N∏j=1

f (ξk, ξ j) · 1ξk (p − qξk)

1∏j�=k(ξ j − ξk)

= pN−1∏j ξ j

− pN−1. (11)

This one is proved by considering the integral

∫ N∏j=1

(p + qzξ j − z) · 1z (p − qz)

· 1∏Nj=1(ξ j − z)

dz,

6Doron Zeilberger saw the identity when it was still a conjecture and suggested to the authors thatan identity of I. Schur [5, Problem VII.47] had a similar look about it and might be proved in asimilar way. This led to the proof we present.


over a large circle. The integral, and so the sum of the residues at 0, the ξk, andp/q, equals zero. This sum is equal to the difference of the two sides of (11).

2.6 The General Particle

The probability PY(xm(t) = x) is the sum of P(X; t) over all X for whichxm = x, thus over all x1, . . . , xm−1 satisfying −∞ < x1 < · · · < xm−1 < x, andall xm+1, . . . , xN satisfying x < xm+1 < · · · < xN < ∞. The latter we can do, asin the last section, since r < 1. Eventually we shall expand the ξσ(i)-contourswhen i < m to CR with R > 1 so that we can sum over these xi.

First take a partition (S−, S+) of [1, N] with |S−| = m − 1 and sum overall those σ for which σ([1, m − 1]) = S− and σ([m, N]) = S+. (At the end wewill sum over these partitions.) Set σ− = σ |[1, m−1], σ+ = σ |[m, N]. Then σ− maybe associated in an obvious way with a permutation in Sm−1 and σ+ with apermutation in SN−m. In particular, sgn σ± make sense, and counting inversionsshows that

sgn σ = (−1)κ(S−, S+) sgn σ− sgn σ+,

where we define in general7

κ(U, V) = #{(i, j) : i ∈ U, j ∈ V, i � j}.When we write ∏

i

ξxiσ(i) =

∏i<m

ξxiσ(i)

∏i�m

ξxiσ(i), (12)

the σ in the first product on the right may be replaced by σ− and the σ in thesecond product on the right may be replaced by σ+.

Similarly, may rewrite (4) as∏

i< j<m

f (ξσ(i), ξσ( j))∏

i∈S−, j∈S+

f (ξi, ξ j)∏

m�i< j

f (ξσ(i), ξσ( j))

∏i< j

f (ξi, ξ j), (13)

and the σ in the first product may be replaced by σ− and the σ in the lastproduct may be replaced by σ+.

When we sum∏

i�m ξxiσ(i) over the xi with i � m (which we may do since

r < 1) we obtain

ξσ(m+1)ξ2σ(m+2) · · · ξ N−1

σ(N)

(1 − ξσ(m+1)ξσ(m+2) · · · ξσ(N))(1 − ξσ(m+1) · · · ξσ(N)) · · · (1 − ξσ(N))

∏i�m

ξ xσ(i).

7In the cited papers we used the notation σ(U, V).


We then multiply by the last factor in the numerator in (13) times sgn σ+ andsum over σ+. The result is, by (10),

p(N−m)(N−m+1)/2

(1 −

∏i∈S+

ξi

) ∏i< j

i, j∈S+

(ξ j − ξi)

∏i∈S+

(1 − ξi)

∏i∈S+

ξ xi . (14)

What remains from (12) and (13) is∏i< j<m

f (ξσ(i), ξσ( j))∏

i∈S−, j∈S+

f (ξi, ξ j)

∏i< j

f (ξi, ξ j)

∏i<m

ξxiσ(i). (15)

The next step is to expand all contours to CR with R > 1, so that we cansum over the xi with i < m. When we expand the contours we encounter polesfrom the denominators in (15) and (14), and integrating the residues wouldgive lower-dimensional integrals. In fact the lower-dimensional integrals from(15) are zero and the lower-dimensional integrals from (14) are of the sametype but with fewer variables. Let us see why this is so. We assume p, q �= 0.

We first consider the poles coming from the denominator in (15) and beginby expanding the ξN contour to a circle CR with R very large. From thedenominator in (4) we encounter poles at

ξN = ξi − pqξi

,

with i < N. As in the proof of Lemma 1 we find that the residue at this poleis analytic for ξi inside Cr, so the integral with respect to ξi equals zero. (Animportant point is that the term ε(ξi) + ε(ξN), which appears in the exponentialin the integrand, becomes analytic at ξi = 0 after the substitution.) Afterexpanding the ξN contour to CR we expand the ξN−1-contour. Now from thedenominator in (15) we have poles at

ξN−1 = ξi − pqξi

,

with i < N − 1. As before, the integral with respect to ξi of the residue is equalto zero. There is also the pole at

ξN−1 = p1 − q ξN

.

But ξN ∈ CR, and if R is chosen large enough this pole is inside Cr and so is notcrossed in the expansion.

Continuing, we find that when we expand all the contours the poles ofAσ do not contribute. But the poles of (14) do contribute, and we get asum of lower-dimensional integrals, one for each subset S′+ ⊂ S+. These areminus the integrals of the residues at the ξk = 1 with k ∈ S+\S′+. If we use


f (ξk, 1) = p (1 − ξk) and f (1, ξk) = q (ξk − 1) we find that this residue is aconstant involving powers of p and q times the same integrand we had beforeexcept that there are no terms involving the ξk with k ∈ S+\S′+.

Once all contours are CR (here R > 1 should be so large that all poles ofthe Aσ lie inside CR) we may sum over all x1, . . . , xm−1 statisfying −∞ < x1 <

· · · < xm−1 < x. The result of this sum is that the product∏

i<m ξxiσ(i) in (15) is

replaced by

1(ξσ(1) − 1)(ξσ(1)ξσ(2) − 1) · · · (ξσ(1) · · · ξσ(m−1) − 1)

∏i∈S−

ξ xi .

Now we are to multiply by sgn σ− and sum over all σ−. An identity8 analogousto (10) tells us that the sum equals

q(m−1)(m−2)/2

∏i< j

i, j∈S−

(ξ j − ξi)

∏i∈S−

(1 − ξi)

∏i∈S−

ξ xi .

We now put everything together. We use the notations

τ = p/q, κ(U) = κ(U, Z+) = sum of the elements of U.

The result (a special case of [12, Theorem 3]) is that when q �= 0,

PY(xm(t) = x) =∑

S−, S′+

(−1)m−1 τ κ(S−∪S′+)−m(m−1)/2−mk qk(k−1)/2+(m−1)(m−2)/2

×∫CR

· · ·∫CR

∏i∈S−, j∈S′+

f (ξi, ξ j)

∏i< j

f (ξi, ξ j)

(1 −

∏j∈S′+

ξ j

)

∏i∈S−, j∈S′+

(ξ j − ξi)

×

∏i< j

(ξ j − ξi)

∏i

(1 − ξ j)

∏i

(ξx−yi−1i eε(ξi) t)

∏i

dξi.

Here k = |S′+| and the sum runs over all disjoint sets S− and S′+ with |S−| =m − 1; indices not otherwise specified run over S− ∪ S′+.

There is one more step. We take a fixed set S and first sum over all partitions(S−, S′+) of S with |S−| = m − 1. (At the end we will sum over all these S). Theonly terms that involve S− and S′+ individually combine as

∏i∈S−, j∈Sc−

f (ξi, ξ j)

ξ j − ξi

(1 −

∏j∈Sc−

ξ j

),

8Proved by interchanging p and q in (10) and letting ξi → 1/ξm−i.


where Sc− denotes the complement of S− in S. Identity (1.9) of [9] (with slightlydifferent notation) is

∑|S−|=m−1

S−⊂S


f (ξi, ξ j)

ξ j − ξi

(1 −

∏j∈Sc−

ξ j

)= qm−1

[ |S| − 1m − 1

]τ

(1 −

∏i∈S

ξi

), (16)

where the τ -binomial coef f icient[

Nn

]τ

is defined by

[Nn

]τ

= (1 − τ N) (1 − τ N−1) · · · (1 − τ N−n+1)

(1 − τ) (1 − τ 2) · · · (1 − τ n).

Hence, after some algebra, the result becomes

Theorem [9, Theorem 5.2] We have when q �= 0,

PY(xm(t) = x) = (−1)m−1 τm(m−1)/2∑

|S|�m

τ κ(S)−mk qk(k−1)/2[

k − 1m − 1

]τ

×∫Ck

R

∏i< j

ξ j − ξi

f (ξi, ξ j)

1 − ∏i ξi∏

i(1 − ξi)

∏i

(ξx−yi−1i eε(ξi) t) dkξ, (17)

where now k = |S| and all indices in the integrand run over S.

Identity (16) depends on a simpler identity,

∑|S−|=m−1

S−⊂S


f (ξi, ξ j)

ξ j − ξi=

[ |S|m − 1

]τ

. (18)

This is proved by induction |S|. We first observe that the left side is apolynomial in the ξi. The reason9 is that it is symmetric in the ξi, and if wemultiply it by the Vandermonde

∏i< j(ξi − ξ j) we obtain an antisymmetric

polynomial which is, therefore, a polynomial times the Vandermonde. Sincethe left side it is bounded as each ξi → ∞ it is a constant. Using the inductionhypothesis and a recursion formula for the τ -binomial coefficients we see bysetting some ξi = 1 that the two sides of the identity agree.

To prove (16) by induction, we see now that the left side is a polynomialof degree at most one in each ξi and, using the induction hypothesis and therecursion formula for the τ -binomial coefficients, that the two sides agree whenany ξi = 1. Therefore the difference is of the form c

∏i(ξi − 1). To show that

9We learned this argument from Anne Schilling.


c = 0 we use (18) to see that after dividing by some ξi the two sides of (16) havethe same limit as ξi → ∞.

3 Fredholm Determinant Representation for Step Initial Condition

3.1 Series Representation

Until now we assumed a system of finitely many particles. Because we can takearbitrarily large R in (17) the result extends to initial configurations

y1 < y2 < . . . → +∞,

where the sum is taken over all finite sets S ⊂ Z+.

For step initial configuration, where Y = Z+ and yi = i, we may sum over

all sets S with |S| = k and so obtain instead a sum over k � m. Before that,instead of indexing the variables in the integrand by S we index them by{1, . . . , k}, so that we can sum under the integral signs for all S with |S| = k. IfS = {s1, . . . , sk} with s1 < · · · < sk then after renaming the variables the factor∏

i ξ−yii in the integrand becomes

∏i ξ

−sii and τ κ(S) becomes

∏i τ

si . These arethe only terms in (17) that involve the individual si. Summing the product ofthese two over all si satisfying 1 � s1 < · · · < sk < ∞ gives, when R > τ ,

1((ξ1/τ)(ξ2/τ) · · · (ξk/τ) − 1

) ((ξ2/τ) · · · (ξk/τ) − 1

)· · ·

((ξk/τ) − 1

) . (19)

The factor∏

i< j f (ξi, ξ j)−1 in the integrand may be written

∏i> j

f (ξi, ξ j)

∏i �= j

f (ξi, ξ j).

If we multiply (19) by the numerator here the rest of the integrand isantisymmetric in the ξi. Thus the integral is unchanged if this product isantisymmetrized. We make the substitution ξi → τ/ξi, use identity (10) withp and q interchanged, and find that the antisymmetrization is

1k! pk(k+1)/2

∏i> j

(ξ j − ξi)

∏i

(qξi − p).

Thus we obtain,


Theorem [9, Corollary to Th. 5.2] For step initial condition we have whenq �= 0,

P(xm(t) � x) = (−1)m∑k�m

1k!

[k − 1k − m

]τ

τm(m−1)/2−mk+k/2 (pq)k2/2

×∫CR

· · ·∫CR

∏i �= j

ξ j − ξi

f (ξi, ξ j)

∏i

ξ xi eε(ξi)t

(1 − ξi) (qξi − p)dξ1 · · · dξk.

Notice that on the left side we have P(xm(t) � x) rather than P(xm(t) = x)

and on the right side the sign is different and the factor 1 − ∏i ξi is gone. This

is the result of summing the formula for P(xm(t) = x) from −∞ to x.For TASEP with p = 0 only the term k = m is nonzero, the multiple integral

is an m × m Toeplitz determinant, and we get

P(xm(t) � x) = det(∫

CR

ξ i− j+x−1 (ξ − 1)−m e(ξ−1)t dξ

).

This was obtained by Rákos and Schütz [6] who showed it was equivalent toJohansson’s result mentioned in the introduction.

3.2 Fredholm Determinant Representation

If we make the change of variables

ξi = 1 − τηi

1 − ηi,

then

∏i �= j

ξ j − ξi

p + qξiξ j − ξi= (1 + τ)k(k−1)

∏i �= j

ηi − η j

τηi − η j.

The right side can be represented in terms of the Cauchy determinant

det(

1τηi − η j

)= τ k(k−1)/2

∏i �= j(ηi − η j)∏i, j(τηi − η j)

.

Going back to the ξi gives the identity

∏i �= j

ξ j − ξi

p + qξiξ j − ξi= (−1)k (pq)−k(k−1)/2

∏i

(1 − ξi)(qξi − p)

· det(

1p + qξiξ j − ξi

)1�i, j�k

.


The theorem becomes

P(xm(t) � x) = (−1)m τm(m−1)/2∑k�m

[k − 1k − m

]τ

τ (1−m)k

× (−1)k

k!∫CR

· · ·∫CR

det(K(ξi, ξ j))1�i, j�k dξ1 · · · dξk,

where

K(ξ, ξ ′) = qξ x eε(ξ)t

p + qξξ ′ − ξ.

Denote by K the operator acting on functions on CR by

K f (ξ) =∫CR

K(ξ, ξ ′) f (ξ ′) dξ ′.

The Fredholm expansion is

det(I − λK) =∞∑

k=0

(−λ)k

k!∫CR

· · ·∫CR

det(K(ξi, ξ j))1�i, j�k dξ1 · · · dξk,

which gives

(−1)k

k!∫CR

· · ·∫CR

det(K(ξi, ξ j))1�i, j�k dξ1 · · · dξk =∫

det(I − λK)

λk+1 dλ,

where we take any contour enclosing λ = 0. Thus,

P(xm(t) � x) = (−1)m τm(m−1)/2∑k�m

[k − 1k − m

]τ

τ (1−m)k∫

det(I − λK)

λk+1 dλ.

If the contour is Cρ with ρ > τ 1−m then we can interchange the sum and integraland use the τ -binomial theorem

∑k�m

[k − 1k − m

]τ

zk =m∏

j=1

z1 − τm− jz

with z = τ 1−m λ−1. We obtain,

Theorem [10, Formula (1)] We have when q �= 0,

P (xm(t) � x) =∫

det(I − λK)∏m−1k=0 (1 − λ τ k)

dλ

λ, (20)

where the contour of integration encloses all the singularities of the integrand.

We can evaluate the integral by residues, getting a finite sum of determi-nants. When m = 1 we obtain

P (x1(t) > x) = det(I − K).


4 Asymptotics

4.1 Statements of the Results

If q > p and t → ∞, we expect xm(t) to be large and negative. We obtain threeasymptotic results for P (xm(t) � x) as t → ∞. Recall the definition γ = q − p.

Theorem 1 [11, Theorem 1] Let m and x be f ixed with x < m. Then as t → ∞

P (xm(t/γ ) > x) ∼∞∏

k=1

(1 − τ k)t2m−x−2 e−t

(m − 1)! (m − x − 1)! .

Theorem 2 [11, Theorem 2] For f ixed m we have

limt→∞ P

(xm(t/γ ) � −t + γ 1/2 s t1/2) =

∫det(I − λKs)∏m−1k=0 (1 − λ τ k)

dλ

λ,

where Ks is the operator on L2(s, ∞) with kernel

K(z, z′) = q√2π

e−(p2+q2) (z2+z′2)/4+pq zz′.

For the third result we recall that

F2(s) = det(I − KAiry χ(s,∞)

),

where

KAiry(x, y) =∫ ∞

0Ai(x + z) Ai(y + z) dz.

For σ ∈ (0, 1) we set

c1(σ ) = −1 + 2√

σ , c2(σ ) = σ−1/6 (1 − √σ)2/3. (21)

Theorem 3 [11, Theorem 3] We have

limt→∞ P

(x[σ t](t/γ ) � c1(σ ) t + c2(σ ) s t1/3) = F2(s)

uniformly for σ in compact subsets of (0, 1).

For TASEP (p = 0) the probabilities are m × m determinants. For m andx fixed the asymptotics of the determinant are easily found and agree withTheorem 1.

A special case of Theorem 2 is

limt→∞ P

(x1(t/γ ) > −t − γ 1/2 s t1/2) = det(I − Ks).

This is a family of distribition functions parameterized by p ∈ [0, 1). Whenp = 0 it is a normal distribution and the probability on the left is the probabilityfor a free particle.


Theorem 3 when p = 0 gives the asymptotics for TASEP obtained byJohansson [2]. A consequence of Theorem 3 is that for fixed s ∈ (0, 1)

limt→∞

x[σ t](t/γ )

t= c1(σ ).

in probability. In fact Liggett [4] showed that this holds almost surely.

4.2 Preliminaries

A natural approach to the asymptotics is to look for a limiting operator K∞such that det(I − λK) → det(I − λK∞) as t → ∞. Once one has guessed K∞there are two possible approaches:

(i) Show that K → K∞ in trace norm.(ii) Show that tr Kn → tr Kn∞ for each n ∈ Z

+ and that K is bounded in tracenorm (or even Hilbert-Schmidt norm) as t → ∞. This suffices because ofthe general formula

log det(I − λL) = −∞∑

n=1

λn

ntr Ln, (22)

which holds for sufficiently small λ.

Both approaches will be used eventually. The operators K on CR haveexponentially large norms as t → ∞, and we will replace them by operatorswith the same Fredholm determinants that are better-behaved. This will bepossible because of lemmas on stability of Fredholm determinants.

Lemma 1 If s → �s is a deformation of closed curves and L(η, η′) is analyticfor η, η′ ∈ �s for all s, then the Fredholm determinant of L acting on �s isindependent of s.

Lemma 2 If L1(η, η′) and L2(η, η′) are two kernels acting on a simple closedcurve �, such that L1(η, η′) extends analytically to η inside � or to η′ inside�, and L2(η, η′) extends analytically to η inside � and to η′ inside �, then theFredholm determinants of L1(η, η′) + L2(η, η′) and L1(η, η′) are equal.

The proofs use the fact that det(I − λL) is determined by the traces tr Ln.For Lemma 1 we use

tr Ln =∫

�s

· · ·∫

�s

L(η1, η2) · · · L(ηn−1, ηn) L(ηn, η1) dη1 · · · dηn.

By analyticity the integral is invariant under the deformation. For Lemma 2,we have to show

tr (L1 + L2)n = tr Ln

1 .


If, say, L1(η, η′) extends analytically to η′ inside �, then

L1 L2 (η, η′′) =∫

�

L1(η, η′) L2(η′, η′′) dη′ = 0.

Similarly L22 = 0. Also, tr L2 = 0, so tr (L1 + L2) = tr L1. Since L1L2 = L2

2 =0, we have for n > 1

(L1 + L2)n = Ln

1 + L2 Ln−11 .

Since

tr L2 Ln−11 = tr Ln−1

1 L2 = 0,

we have tr (L1 + L2)n = tr Ln

1 .

4.3 Another Operator

We introduce the notation

ϕ(η) =(

1 − τη

1 − η

)x

e[

11−η

− 11−τη

]t.

In K(ξ, ξ ′) we make the substitutions

ξ = 1 − τη

1 − η, ξ ′ = 1 − τη′

1 − η′ , t → t/γ,

and obtain the kernel10

ϕ(η′)η′ − τη

= K2(η, η′),

acting on c, a little circle about η = 1 described clockwise, which has the sameFredholm determinant. We denote this by K2 because there is an equallyimportant kernel

ϕ(τη)

η′ − τη= K1(η, η′).

The kernel K1(η, η′) extends analytically to η and η′ inside c while K2(η, η′)extends analytically to η inside c. Hence by Lemma 2 the determinant of K2equals the determinant of K2 − K1.

Next we apply Lemma 1 to the kernel

K1(η, η′) − K2(η, η′) = ϕ(τη) − ϕ(η′)η′ − τη

,

with �0 = −c and �1 = Cρηwith 1 < ρη < τ−1. (Recall that c was described

clockwise.) Since the numerator vanishes when the denominator does, the onlysingularities of the kernel are at η, η′ = 1, τ−1, neither of which is passed in

10This is the kernel (dξ/dη)1/2(dξ ′/dη′)1/2 K(ξ(η), ξ ′(η′)).


a deformation �s, s ∈ [0, 1]. Therefore the operator K acting on CR may bereplaced by K1 − K2 acting on Cρη

.

4.4 Another Fredholm Determinant Representation

The function ϕ(τη) is analytic on sCρ when 0 < s � 1. The denominator η′ −τη in K1 is nonzero for η, η′ ∈ sCρ for all such s. Therefore by Lemma 1 theFredholm determinant of K1 on Cρ is the same as on sCρ . This in turn is thesame as the Fredholm determinant of

s K1(sη, sη′) = ϕ(sτη)

η′ − τη, (23)

on Cρ . As s → 0 this converges in trace norm to the kernel

K0(η, η′) = 1η′ − τη

,

on Cρ . Therefore the Fredholm determinant of K1 equals the Fredholmdeterminant of K0.

The kernel of K20 equals

K20(η, η′) =

∫Cρ

dζ

(ζ − τη) (η′ − τζ )= 1

η′ − τ 2 η,

because τη is inside Cρ and τ−1η′ outside when η, η′ ∈ Cρ . In particular tr K20 =

(1 − τ 2)−1. Generally, we find that tr Kn0 = (1 − τ 2)−n. Thus by (22) we have

for small λ

log det(I − λK0) = −∞∑

n=1

λn

n1

1 − τ n= −

∞∑k=0

∞∑n=1

τ nkλn

n=

∞∑k=0

log(1 − λτ k),

and so

det(I − λK1) = det(I − λK0) =∞∏

k=0

(1 − λτ k).

We factor out I − λK1 in

P(xm(t/γ ) � x) =∫

det(I − λK)∏m−1k=0 (1 − λ τ k)

dλ

λ=

∫det(I − λK1 + λK2)∏m−1

k=0 (1 − λ τ k)

dλ

λ,

(recall the substitution t → t/γ ) and obtain

P(xm(t/γ ) � x) =∫ ∞∏

k=m

(1 − λ τ k) det(I + λK2 (I + R))dλ

λ, (24)

where R is the resolvent operator λK1 (I − λK1)−1.


4.5 Theorems 1 and 2

Consecutive integration shows that for small λ the resolvent kernel has the nicerepresentation

R(η, η′; λ) =∞∑

n=1

λn ϕn(τη)

η′ − τ nη, (25)

where

ϕn(η) = ϕ(η) ϕ(τη) · · · ϕ(τ n−1η).

For Theorems 1 and 2, whose derivations we shall not explain in detail, wewrote R = R1 + R2 where R1 is analytic everywhere except for the poles atλ = 1, τ−1, . . . , τ−m+1 and R2 is analytic for |λ| < τ−m. For Theorem 1 theasymptotics comes from the residue of R1 at λ = τ−m+1. For Theorem 2 weused approach (ii) described above. In [10] a steepest descent computation hadshown that tr Kn → tr Kn for all n. What was needed to complete the proofwas to show that K2 (I + R) had bounded Hilbert-Schmidt norm as t → ∞,uniformly for λ in compact sets not containing any of the singularities τ−k. Weused the representation R = R1 + R2 to show that this was so.

4.6 Theorem 3

Here m = σ t is large and 1, τ−1, . . . , τ−m+1 must be inside the contour. If weset λ = μτ−m we can take μ ∈ Cρ with ρ > τ fixed, and (24) becomes

P(xm(t/γ ) � x) =∫ ∞∏

k=0

(1 − μτ k) det(I + μτ−m K2 (I + R))dμ

μ. (26)

In (25) we use

ϕn(η) = ϕ∞(η)

ϕ∞(τ n η),

where

ϕ∞(η) = limn→∞ ϕn(η) = (1 − η)−x e

η

1−ηt.

The Cauchy integral representation of ϕ∞(τ n η)−1, and some manipulation ofseries and integrals, give

K2(I + R) (η, η′) = −∫

|ζ |>ρη

ϕ(ζ )

(ζ − τη) (η′ − ζ )dζ +

∞∑k=−∞

τ k

1 − τ kλ

×∫Cρζ

ϕ∞(ζ )

ζ − τηζ k dζ

∫Cρu

1ϕ∞(u) (η′ − u/τ)

duuk+1 ,


where the radii of the contours in the series satisfy

ρζ ∈ (1, ρη), ρu ∈ (τ ρζ , τ ρη).

The first operator on the right side is analytic for |η|, |η′| � ρη and theinfinite sum is analytic for |η| � ρη. It follows by Lemma 2 that the Fredholmdeterminant of the sum of the two, i.e., of K2(I + R), equals the Fredholmdeterminant of the infinite sum.

If we set

f (μ, z) =∞∑

k=−∞

τ k

1 − τ kμzk,

then since λ = μτ−m,∞∑

k=−∞

τ k

1 − τ kλ

(ζ

u

)k

= τm(

ζ

u

)m

f (μ, ζ/u),

and so the infinite sum may be written

τm∫Cρu

∫Cρζ

ϕ∞(ζ )

ϕ∞(u)

(ζ

u

)m f (μ, ζ/u)

(ζ − τη) (η′ − u/τ)dζ

duu

.

The substitutions η → η/τ, η′ → η′/τ replace this by the kernel

τm∫Cρu

∫Cρζ

ϕ∞(ζ )

ϕ∞(u)

(ζ

u

)m f (μ, ζ/u)

(ζ − η) (η′ − u)dζ

duu

,

where now the operator acts on Cρηwith ρη ∈ (τ, 1) and in the integral

ρζ ∈ (1, τ−1 ρη), ρu ∈ (τρζ , ρη).

If we expand the u-integral so that ρη < |u| < 1 on the new contour we passthe pole at u = η′, which gives the contribution

τm∫Cρζ

ϕ∞(ζ )

ϕ∞(η′)ζ m

(η′)m+1

f (μ, ζ/η′)ζ − η

dζ. (27)

The new double integral is analytic for |η|, |η′| � ρη and (27) is analytic for|η| � ρη. Therefore by Lemma 2 the Fredholm determinant is the same as thatof (27).

We have shown that if we define

J(η, η′) =∫Cρζ

ϕ∞(ζ )

ϕ∞(η′)ζ m

(η′)m+1

f (μ, ζ/η′)ζ − η

dζ, (28)

where ρζ ∈ (1, τ−1 ρη), then (26) becomes

P(xm(t/γ ) � x) =∫ ∞∏

k=0

(1 − μτ k) det(I + μ J)dμ

μ. (29)


This representation, in which the parameter m is in the operator, makes anasymptotic analysis possible.

By Lemma 1 the contours Cρη(the home of the functions on which J acts)

and Cρζ(in the integral defining J) may be simultaneously deformed if during

the deformation we do not pass a singularity of the integrand.We apply steepest descent, and so look for the saddle points for ϕ(ζ ) ζ m

when m ∼ σ t and x ∼ c t. In general there are two saddle points. When c equalsc1(σ ), given in (21), they coincide at

ξ = −√σ/(1 − √

σ).

Both contours may be deformed to pass through the saddle point, the neigh-borhood of which gives the main contributions. If x = c1(σ ) t + c2(σ ) s t1/3

precisely, and we make the variable changes

η → ξ + t−1/3 c3 η, η′ → ξ + t−1/3 c3 η′, ζ → ξ + t−1/3 c3 ζ

for a certain constant c3, then the rescaled kernel μ J(μ, μ′) has limit∫

�ζ

e−ζ 3/3+sζ+(η′)3/3−sη′

(ζ − η) (η′ − ζ )dζ.

(The constants c2(σ ) and c3 come from a third derivative at the saddle point.)Here �ζ consists of the the rays from 0 to ∞ e±2π i/3. The limiting operator actson functions on the contour �η consisting of the the rays from 0 to ∞ e±π i/3.

Using the general identity det(I − AB) = det(I − BA) we replace this bythe kernel ∫

�ζ

∫�η

e−ζ 3/3+η3/3+yζ−xη

ζ − ηdη dζ = −KAiry(x, y),

acting on L2(s, ∞), where

KAiry(x, y) =∫ ∞

0Ai(z + x) Ai(z + y) dz.11

Hence

det(I + μ J) → det(I − KAiry χ(s, ∞)

) = F2(s)

for all μ, and it follows that the integral in (29) has the limit F2(s).

Acknowledgements The second author thanks the Université de Paris 7 for their invitation,hospitality, and generous support during his visit in June, 2009.

The authors were supported by the National Science Foundation through grants DMS-0906387(first author) and DMS-0854934 (second author).

11The reason the double integral equals −KAiry(x, y) is that applying the operator ∂/∂x + ∂/∂y tothe two kernels gives the same result, Ai(x) Ai(y), so they differ by a function of x − y. Since bothkernels go to zero as x and y go to +∞ independently this function must be zero.


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3. Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett.56, 889–892 (1986)

4. Liggett, T.M.: Interacting Particle Systems. Springer, Berlin (2005). Reprint of the 1985original

5. Pólya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis. Springer, Berlin (1964)6. Rákos, A., Schütz, G.M.: Current distribution and random matrix ensembles for an integrable

asymmetric fragmentation process. J. Stat. Phys. 118, 511–530 (2005)7. Schütz, G.M.: Exact solution of the master equation for the asymmetric exclusion process. J.

Stat. Phys. 88, 427–445 (1997)8. Spitzer, F.: Interaction of Markov processes. Adv. Math. 5, 246–290 (1970)9. Tracy, C.A., Widom, H.: Integral formulas for the asymmetric simple exclusion process.

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291–300 (2008)11. Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math.

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process”. Commun. Math. Phys. 304, 875–878 (2011)


Surface Tension of Multi-phase Flow with MultipleJunctions Governed by the Variational Principle

Shigeki Matsutani · Kota Nakano ·Katsuhiko Shinjo

Received: 5 January 2010 / Accepted: 4 July 2011 / Published online: 30 July 2011© Springer Science+Business Media B.V. 2011

Abstract We explore a computational model of an incompressible fluid witha multi-phase field in three-dimensional Euclidean space. By investigating anincompressible fluid with a two-phase field geometrically, we reformulate theexpression of the surface tension for the two-phase field found by Lafaurieet al. (J Comput Phys 113:134–147, 1994) as a variational problem related toan infinite dimensional Lie group, the volume-preserving diffeomorphism. Thevariational principle to the action integral with the surface energy reproducestheir Euler equation of the two-phase field with the surface tension. Since thesurface energy of multiple interfaces even with singularities is not difficult tobe evaluated in general and the variational formulation works for every actionintegral, the new formulation enables us to extend their expression to that ofa multi-phase (N-phase, N � 2) flow and to obtain a novel Euler equationwith the surface tension of the multi-phase field. The obtained Euler equationgoverns the equation for motion of the multi-phase field with different surfacetension coefficients without any difficulties for the singularities at multiplejunctions. In other words, we unify the theory of multi-phase fields which ex-press low dimensional interface geometry and the theory of the incompressiblefluid dynamics on the infinite dimensional geometry as a variational problem.We apply the equation to the contact angle problems at triple junctions. Wecomputed the fluid dynamics for a two-phase field with a wall numericallyand show the numerical computational results that for given surface tensioncoefficients, the contact angles are generated by the surface tension as resultsof balances of the kinematic energy and the surface energy.

S. Matsutani (B) · K. Nakano · K. ShinjoAnalysis Technology Center, Canon Inc. 3-3-20, Shimomaruko, Ota-ku,Tokyo 146-8501, Japane-mail: [email protected]

238 S. Matsutani et al.

Keywords Multi-phase flow · Surface tension · Multiple junction ·Volume-preserving diffeomorphism

Mathematics Subject Classifications (2010) 37K65 · 58E12 · 76T30 · 76B45

1 Introduction

Recently, since the developments of both hardware and software in computerscience enable us to simulate complex physical processes numerically, suchcomputer simulations become more important from industrial viewpoints.Especially the computation of the incompressible multi-phase fluid dynamicshas crucial roles in order to evaluate the behavior of several devices andmaterials in a micro-region, e.g., ink-jet printers, solved toners and so on. Inthe evaluation, it is strongly required that the fluid interfaces with multiplejunctions are stably and naturally computed from these practical reasons.

In this article, in order to handle the fluid interfaces with multiple junctionsin a three dimensional micro-region, we investigate a surface tension ofan incompressible multi-phase flow with multiple junctions as a numericalcomputational method under the assumption that the Reynolds number isnot so large. In the investigation, we encounter many interesting mathemat-ical objects and results, which are associated with low dimensional interfacegeometry having singularities, and with the infinite dimensional geometry ofincompressible fluid dynamics. Further since even in a macroscopic theory,we introduce artificial intermediate regions in the material interfaces amongdifferent fluids or among a solid and fluids, the regions give a resolution of thesingularities in the interfaces to provide extended Euler equations naturally.Thus even though we consider the multi-phase fluid model as a computationalmodel, we believe that it must be connected with mathematical nature of realfluid phenomena as their description. We will mention the background, themotivation and the strategy of this study more precisely as follows.

For a couple of decades, in order to represent the physical process withthe interfaces of the multi-phase fluids, the computational schemes have beenstudied well. These schemes are mainly classified into two types. The first typeis based on the level-set method [41] discovered by Zhao et al. [48, 49]. Thesecond one is based on the phase-field theory, which was found by Brakbillet al. [9], and Lafaurie et al. [30]. The authors in [30] called the schemeSURFER. Following them, there are many studies on the SURFER scheme,e.g., [7, 12, 25, references therein].

The level-set method is a computational method in which we describe a(hyper-)surface in terms of zeros of the level-set function, i.e., a real functionwhose value is a signed distance from the surface, such as q(x) in Section 2.1.Using the scheme based upon the level-set method in the three dimensionalEuclidean space, we can deal well with topology changes, geometrical objectswith singularities, e.g., cusps, the multiple junctions of materials, and so on.

Surface Tension of Multi-phase Flow 239

However in the computation, we need to deal with the constraint conditionseven for two-phase fluids [48, 49]. A dynamical problem with constraint condi-tions is basically complicate and sometimes gives difficulties to find its solutionsince the constraint conditions sometimes generate an ill-posed problem in theoptimization. In the numerical computation for incompressible fluid, we mustcheck the consistency between the incompressible condition and the constraintcondition. The check generally requires a complicate implementation of thealgorithm, and increases computational cost. Its failure sometimes makes thecomputation unstable, especially when we add some other physical conditions.Since instability disturbs the evaluation of a complex system as a model of areal device, it must be avoided.

On the other hand, using the SURFER scheme [30], we can easily com-pute effects of the surface tension of a two-phase fluid in the Navier-Stokesequation. The phase field model is the model that we represent materialsin terms of supports of smooth functions which roughly correspond to thepartition of unity in pure mathematics [28, I p.272] as will be mentioned inSections 4 and 5. We call these functions “color functions” or “phase fields”.The phase fields have artificial intermediate regions which represent theirinterfaces approximately. In the SURFER scheme [30], the surface tensionis given as a kind of stress force, or volume force due to the intermediateregion. Hence the scheme makes the numerical computations of the surfacetension stable. However it is not known how to consider a multi-phase(N-phase, N � 2) flow in their scheme. In [9], the authors propose a methodas an extension of the SURFER scheme [30] to the contact angle problemby imposing a constraint to fix its angle. In this article, we will generalize theSURFER scheme to multi-phase flow without any constraints.

Nature must not impose any constraints even at such a triple junction,which is governed by a physical principle. If it is a Hamiltonian system, itsdetermination must obey the minimal principle or the variational principle.We wish to find a theoretical framework in which we can consistently handlethe incompressible flows with interfaces including the surface tensions and themultiple junctions without any constraints. As the multiple junctions should betreated as singularities in a mathematical framework which are very difficultto be handled in general, it is hard to extend mathematical approaches forfluid interface problems without a multiple junction [8, 42] to a theory for theproblem with multiple junctions. Our purpose of this article is to find such atheoretical framework which enables us to solve the fluid interface problemswith multiple junctions numerically as an extension of the SURFER scheme.

For the purpose, we employ the phase field model. The thickness ofthe actual intermediate region in the interface between a solid and a fluidor between two fluids is of atomic order and is basically negligible in themacroscopic theory. However the difference between zero and “the limit tozero” sometimes brings a crucial difference in physics and mathematics; forexample, in the Sato hyperfunction theory, the delta function is regardedas a function in the boundary of the holomorphic functions [23, 27], i.e.,


δ(x) = limε→0

12π i

( 1x−iε − 1

x+iε

) ≡ limε→0

1π

εx2+ε2 . As mentioned above, the phase field

model has the artificial intermediate region which is controlled by a smallparameter ε and appears explicitly even as a macroscopic theory. We regardthat it represents the effects coming from the actual intermediate region ofmaterials. Namely, we regard that the stress force expression in the SURFERscheme is caused by the artificial intermediate region of the phase-fields and itrepresents well the surface effect coming from that of real materials.

In order to extend the stress force expression of the two-phase flow tothat of the multi-phase (N-phase, N � 2) flow, we will first reformulate theSURFER scheme in the framework of the variational theory. In [25], a similarattempt was reported but unfortunately there were not precise derivations.Our investigations in Section 4 show that the surface tension expression ofthe SURFER scheme is derived as a momentum conservation in Noether’stheorem [11, 24] and its derivation requires a generalization of the Laplaceequation [31] as the Euler–Lagrange equation [1, 11], which is not trivial evenfor a static case.

In order to deal with this problem in a dynamics case consistently, weshould also consider the Euler equation in the framework of the variationalprinciple. It is well-known that the incompressible fluid dynamics is geomet-rically interpreted as a variational problem of an infinite dimensional Liegroup, related to diffeomorphism, due to Arnold [4, 5], Ebin and Marsden[16], Omori [38] and so on. Following them, there are so many related works[2, 10, 26, 37, 40, 43, 44, 46].

On the reformulation of the SURFER scheme [30] for the dynamical case,we introduce an action integral including the kinematic energy of the incom-pressible fluid and the surface energy. The variational method reproduces thegoverning equation in the SURFER scheme.

After then, we extend the surface energy to that of multi-phase fieldsand add the energy term to the action integral. The variational principle ofthe action integral leads us to a novel expression of the surface tension andthe extended Euler equation which we require. Using the extended Eulerequation, we can deal with the surface tensions of the multi-phase flows, themultiple junctions of the of phase fields including singularities, the topologychanges and so on. We can also compute a wall effect naturally and a contactangle problem. The computation of the governing equation is freed from anyconstraints, except the incompressible condition.

In other words, in this article, we completely unify the theory of the multi-phase (N-phase, N � 2) field and the theory of the incompressible fluiddynamics of Euler equation as an infinite dimensional geometrical problem.

Contents are as follows: Section 2 is devoted to the preliminaries of thetheory of surfaces in our Euclidean space from a low-dimensional differentialgeometrical viewpoint [19, 20, 34] and Noether’s theorem in the classical fieldtheory [1, 11, 24]. Section 3 reviews the derivation of the Euler equation on theincompressible fluid dynamics following the variational method for an infinite-dimensional Lie algebra based upon [16]. In Section 4, we reformulate theSURFER scheme [30]. There the Laplace equation for the surface tension and


the Euler equation in [30] are naturally obtained by the variational methodin Propositions 8 and 10. Section 5 is our main section in which we extend thetheory in [30] to that for a multi-phase flow and obtain the Euler equation withthe surface tension of the multi-phase field in Theorem 2. The extended Eulerequation for the multi-phase flow is derived from the variational principle ofthe action integral in Theorem 1. As a special case, we also derive the Eulerequation for a two-phase field with wall effects in Theorem 3. In Section 6,using these methods in the computational fluid dynamics [15, 21, 22], weconsider numerical computations of the contact angle problem of a two-phasefield because the contact angle problem for the two-phase field circumscribedin a wall is the simplest non-trivial triple junction problem. By means of ourscheme, for given surface tension coefficients, we show two examples of thenumerical computations in which the contact angles automatically appearedwithout any geometrical constraints and any difficulties for the singularitiesat triple junctions. The computations were very stable. Precisely speaking,as far as we computed, the computations did not collapse for any boundaryconditions and for any initial conditions.

2 Mathematical Preliminaries

2.1 Preliminary of Surface Theory

In this subsection, we review the theory of surfaces from the viewpoint oflow-dimensional differential geometry. The interface problems have been alsostudied for last three decades in pure mathematics, which are considered as arevision of the classical differential geometry [17] from a modern point of view[18–20, 34, 45], e.g., generalizations of the Weierstrass–Ennpper theory of theminimal surfaces, isothermal surfaces, constant curvature surfaces, constantmean curvature surfaces, Willmore surfaces and so on. They are also closelyconnected with the harmonic map theory and the theory of the variationalprinciple [19, 20].

We consider a smooth surface S embedded in three dimensional Euclideanspace E

3. Let x = (x1, x2, x3) be of the Cartesian coordinate system andrepresent a point in E

3, and let the surface S be locally expressed by a localparameter (s1, s2). We assume that the surface S is expressed by zeros of a realvalued smooth function q over E

3, i.e.,

q(x) = 0,

such that in the region whose |q| is sufficiently small (|q| < εT for a positivenumber εT > 0), |dq| agrees with the infinitesimal length in the Euclideanspace. Then dq means the normal co-vector field (one-form), i.e., for thetangent vector field eα := ∂α := ∂/∂sα (α = 1, 2) of S,

〈∂α, dq〉 = 0 over S = {x ∈ E3 |q(x) = 0}. (2.1)


Here 〈, 〉 means the pointwise pairing between the cotangent bundle and thetangent bundle of E

3. The function q can be locally regarded as so-called thelevel-set function [41, 49]. We could redefine the domain of q such that it isrestricted to a tubular neighborhood TS of S,

TS := {x ∈ E3 | |q(x)| < εT}.

Over TS, q agrees with the level-set function of S. There we can naturallydefine a projection map π : TS → S and then we can regard TS as a fiberbundle over S, which is homeomorphic to the normal bundle NS → S. How-ever the level-set function is defined as a signed distance function which is aglobal function over E

3 as a continuous function [41] and thus it has no naturalprojective structure in general; for example, the level-set function L of a spherewith radius a is given by

L(x1, x2, x3) =√

(x1)2 + (x2)2 + (x3)2 − a,

which induces the natural projective (fiber) structure but the origin (0, 0, 0) inthe sphere case. The level-set function has no projective structure at (0, 0, 0) inthis case, and we can not define its differential there. In other words, the level-set function is not a global function over E

3 as a smooth function in general.When we use the strategy of the fiber bundle and its connection, we restrict

ourselves to consider the function q in TS. Then the relation (2.1) and theparameter (s1, s2) are naturally lifted to TS as an inverse image of π .

Further for eq := ∂q := ∂/∂q, we have

∂α(eq) =∑

β

βαqeβ over S.

Here (βαq) is the Weingarten map, which is a kind of a point-wise 2 × 2-matrix

((βαq)αβ) [28, Chapter VII]. The eigenvalue of (

βαq) is the principal curvature,

whereas a half of its trace tr(βαq)/2 is known as the mean curvature and its

determinant det(βαq) means the Gauss curvature [28, Chapter VII].

Noting the relation, 〈eβ, dsα〉 = δαβ for α, β = 1, 2, the twice of the mean

curvature, κ , is given by,

∑

α

∂α(eq)dsα = κ over S.

Further noting the relation ∂qeqdq = 0, we obtain

∑

α

∂α(eq)dsα + ∂q(eq)dq = κ over S.

Due to the flatness of the Euclidean space, we identify eq with ∇q/|∇q| andthen we have the following proposition.


Proposition 1 The following relation holds at a point over S,

div( ∇q

|∇q|)

= κ.

For the case |∇q| = 1, using the Hodge star operator [1, 36] and the exteriorderivative d, we also have an alternative expression ∗d ∗ dq = κ over thesurface S. Here the Hodge star operator is ∗ : �p(TS) → �3−p(TS) and theexterior derivative d : �p(TS) → �p+1(TS) (dω =∑3

i=1 ∂iωdxi), where �p(TS)

is the set of smooth p-forms over TS [36].Noting that as the left hand side of formula in Proposition 1 can be lifted to

TS, the formula plays an important role in [9, 30, 48] and in this article.

2.2 Preliminary of Noether’s Theorem

In this subsection, we review Noether’s theorem in the variational methodwhich appears in a computation of the energy-momentum tensor-field in theclassical field theory [1, 11, 24].

Let the set of smooth real-valued functions over n-dimensional Euclid-ean space E

n be denoted by C∞(En)⊗ , where n is mainly three. Let x =(x1, x2, . . . , xn) be of the Cartesian coordinate system of E

n. We consider thefunctional I : C∞(En)⊗ → R,

I =∫

EndnxF(φa(x), ∂iφa(x)), (2.2)

where F is a local functional, F : C∞(En)⊗ |x → �n(En)|x,

F : (φa)a=1,..., |x �→F(φa(x), ∂iφa(x))dnx ≡ F(φa(x), ∂1φa(x), . . . , ∂nφa(x))dnx

≡ F(φ1(x), . . . , φ (x), ∂1φ1(x), . . . , ∂nφ (x))dnx

and ∂i := ∂/∂xi, (i = 1, · · · , n). Then we obviously have the the followingproposition.

Proposition 2 For the functional I in Eq. (2.2) over C∞(En)⊗ , the Euler–Lagrange equation coming from the variation with respect to φa of (φb )b=1,..., ∈C∞(En)⊗ , i.e., δI

δφa(x)= 0, is given by

δFδφa(x)

−n∑

i=1

∂iδF

δ∂iφa(x)= 0. (2.3)

Using Eq. (2.3), we consider an effect of a small translation x to x′ = x + δxon the functional I. The following proposition is known as Noether’s theoremwhich plays crucial roles in this article.


Proposition 3 The functional derivative I with respect to δxi is given by

δIδxi

=n∑

j=1

∂ j

[ ∑

a=1

δFδ∂ jφa

∂iφa

]

− ∂i [F] . (2.4)

If I is invariant for the translation, Eq. (2.4) gives the conservation of themomentum.

Proof For the variation x′ = x + δx, the scalar function becomes

φa(x′) = φa(x) +n∑

i=1

∂iφa(x)δxi + O(δx2).

From the relations on the Jacobian and each component,

∂x′

∂x= 1 +

n∑

i=1

∂iδxi + O(δx2),∂xk

∂x′i = δki − ∂iδxk + O(δx2),

we have

∂φa(x′)∂x′i = ∂φa(x) +∑n

j=1 ∂ jφa(x)δx j

∂xk

∂xk

∂x′i + O(δx2)

= ∂iφa +n∑

j=1

(∂i∂ jφa)δx j + O(δx2).

Then up to δx2, we obtain∫

Endnx′F(φa(x′), ∂ ′

i φa(x′)) −∫

EndnxF(φa(x), ∂iφa(x))

=∫

En

⎡

⎣n∑

i=1

∑

a=1

δFδφa

∂iφa(x)δxi +n∑

i, j=1

∑

a=1

δFδ∂ jφa

∂i∂ jφa(x)δxi +n∑

j=1

F∂iδxi

⎤

⎦dnx

=∫

En

⎛

⎝n∑

i=1

∂i

⎡

⎣n∑

j=1

∑

a=1

δFδ∂iφa

∂iφa − F

⎤

⎦ δxi

⎞

⎠dnx.

Here we use the Euler–Lagrange equation (2.3) and then we have Eq. (2.4). Ifwe assume that I is invariant for the variation, it vanishes. �

3 Variational Principle for Incompressible Fluid Dynamics

As we will derive the governing equation as the variational equation of anincompressible multi-phase flow with interfaces using the variational method,let us review the variational theory of the incompressible fluid to obtain theEuler equation following [4, 5, 16, 26, 29, 33, 37].


Let � be a smooth domain in E3. The incompressible fluid dynamics can be

interpreted as a geometrical problem associated with an infinite dimensionalLie group [5, 16, 38]. It is related to the volume-preserving diffeomorphismgroup SDiff(�) as a subgroup of the diffeomorphism group Diff(�). Thediffeomorphism group Diff(�) is generated by a smooth coordinate trans-formation of �. The Lie algebras sdiff(�) ≡ TeSDiff(�) of SDiff(�) anddiff(�) ≡ TeDiff(�) of Diff(�) are the infinite dimensional real vector spaces.The sdiff(�) is a linear subspace of diff(�).

Following Ebin and Marsden [16], we consider the geometrical meaning ofthe action integral of an incompressible fluid,

∫

Tdt∫

�

d3x(

12ρ|u|2)

. (3.1)

Here T := (0, T0) is a subset of the set of real numbers R, (x, t) is the Cartesiancoordinate of the space-time � × T, ρ is the density of the fluid which isconstant in this section, and u = (u1, u2, u3) is the velocity field of the fluid.

Geometrically speaking, a flow obeying the incompressible fluid dynamicsis considered as a section of a principal bundle IFluid(� × T) over the absolutetime axis T ⊂ R as its base space,

SDiff(�) −−−−→ IFluid(� × T)

�

⏐⏐�

T.

(3.2)

The projection � is induced from the trivial fiber structure �� : � × T → T,((x, t) → t). In the classical (non-relativistic) mechanics, every point of space-time has a unique absolute time t ∈ R, which is contrast to one in the relativistictheory.

Due to the Weierstrass polynomial approximation theorem [47], we canlocally approximate a smooth function by a regular function. Let the set ofsmooth functions over � be denoted by C∞(�) and the set of the regular realfunctions by Cω(�) whose element can be expressed by the Taylor expansionin terms of local coordinates.

The action of Diff(�) on Cω(�) ⊂ C∞(�) is given by

esui∂i f (x) = f (x + su),

for an element f ∈ Cω(�), and small s > 0, where ∂i := ∂/∂xi and we use theEinstein convention; when an index i appears twice, we sum over the index i.Thus the action esui∂i is regarded as an element of Diff(�).

As a frame bundle of the principal bundle IFluid(� × T), we consider avector bundle Coor(� × T) with infinite rank,

C∞(�) −−−−→ Coor(� × T)

� ′⏐⏐�

T.


Since C∞(�) is regarded as a non-countably infinite dimensional linear spaceover R, we should regard Diff(�) and SDiff(�) as subgroups of an infinitedimensional general linear group if defined.

More rigorously, we should consider the ILH space (inverse limit of Hilbertspace) (or ILB space (inverse limit of Banach space)) introduced in [38] byadding a certain topology to (a subspace of) C∞(� × T), and then we alsoshould regard Diff and SDiff as an ILH Lie group. However our purpose is toobtain an extended Euler equation from a more practical viewpoint. Thus weformulate the theory primitively even though we give up to consider a generalsolution for a general initial condition.

We consider smooth sections of Coor(� × T) and IFluid(� × T). Smoothsections of Coor(� × T) can be realized as C∞(� × T). In the meaning of theWeierstrass polynomial approximation theorem [47], an appropriate topologyin C∞(� × T) makes Cω(� × T) dense in C∞(� × T) by restricting the region� × T appropriately. Under the assumption, we also deal with a smoothsection of IFluid(� × T).

Let us consider a coordinate function (γ i(x, t))i=1,2,3 ∈ Cω(� × T) such that

ddt

γ i(x, t) = ui(x, t), γ i(x, t) = xi at t ∈ T,

which means

γ i(x, t + δt) = xi + ui(x, t)δt + O(δt2),

for a small δt. Here the addition is given as a Euclidean move in E3. As an

inverse function of γ = γ (u, t), we could regard u as a function of γ and t,

u(x, t) = u(γ (x, t), t).

Further we introduce a small quantity modeled on δt · ui,

γ i(x, t) := γ i(x, t) − xi. (3.3)

Then a section g of IFluid(� × T) at t ∈ T can written by,

g(t) = eγ i∂i ∈ IFluid(� × T)

∣∣∣t≈ SDiff(�) ⊂ Diff(�). (3.4)

Here we consider g as an element of SDiff(�) and thus it satisfies the conditionof the volume preserving, which appears as the constraint that the Jacobian,

∂γ

∂x:= det

(∂γ i

∂x j

)= (1 + tr(∂ jui)δt) + O(δt2),

must preserve 1, i.e., the well-known condition that tr(∂ jui) = div(u) mustvanish, or d

dt∂γ

∂x = 0.Following [16], we reformulate the action integral (3.1) as “the energy

functional” in the frame work of the harmonic map theory. In the harmonic


map theory [20] by considering a smooth map h : M → G for a n-smoothbase manifold M and its target group manifold G, “the energy functional”is given by

E = 12

∫

Mtr((h−1dh) ∗ (h−1dh)

). (3.5)

Here ∗ means the Hodge star operator, which is for ∗ : TG ⊗ �p(M) → TG ⊗�n−p(M) where �p(M) is the set of the smooth p-forms over M [36], andTG ⊗ �p(M) is the set of the tangent bundle TG valued smooth p-forms overM [36]. The term “energy functional” in the harmonic map theory means thatit is an invariance of the system and thus it sometimes differs from an actualenergy in physics.

Since in (3.2), the base space T is dimensional and the target spaceIFluid(� × T)|t at t ∈ T is the infinite dimensional space, “the energy func-tional” (3.5) in the harmonic map theory corresponds to the action integralSfree[γ ] which is defined by

Sfree[γ ] = 12

∫

T

∫

�

∂γ

∂xρd3x · dxi ⊗ dxi

((e−γ k∂k dt

ddt

eγ ∂

)(e−γ j∂ j

ddt

eγ n∂n

)).

Here dxi(∂ j) := 〈∂ j, dxi〉 = δij is the natural pairing between T� and T∗�.

The trace in (3.5) corresponds to the integral over � with ∂γ

∂x ρd3x · dxi ⊗ dxi.

The Hodge ∗ operator acts on the element such as ∗(

e−γ k∂k dt ddt e

γ ∂

)=

(e−γ k∂k d

dt eγ ∂

)as the natural map from diff(�) valued 1-form to 0-form.

Further we assume that ρ is a constant function in this section. Then the actionintegral Sfree[γ ] obviously agrees with (3.1).

We investigate the functional derivative and the variational principle of thisSfree[γ ]. Let us consider the variation,

γ j(x, t′) = γ j(x, t′) + δγ j(x, t′), and γ j(x, t′) = γ j(x, t′) + δγ j(x, t′),

where we implicitly assume that δγ j is proportional to the Dirac δ function,δ(t′ − t), for some t and δγ j vanishes at ∂�. As we have concerns only for localeffects or differential equations, we implicitly assume that we can neglect theboundary effect arising from ∂� on the variational equation. If one needs theboundary effect, he would follow the study of Shkoller [43]. Further one coulduse the language of the sheaf theory to describe the local effects [27]. As we areconcerned only with differential equation and thus our theory is completelylocal except Section 6, we could deal with germs of related bundles [6] as in[34], which is also naturally connected with a computational method of fluiddynamics [35].

Let us consider the extremal point of the action integral Eq. (3.1) followingthe variational principle. Noting that ∂γ /∂x = 1, the above Jacobian becomes

∂(γ + δγ )

∂x= ∂γ

∂x

(1 + ∂kδγ

k)+ O((δγ )2).


Since we employ the projection method, we firstly consider a variation indiff(�) rather than sdiff(�). For the variation, the action integral Sfree[γ ] with(3.4) becomes

Sfree[γ + δγ ] − Sfree[γ ]

= −∫

Tdt∫

�

∂γ

∂xd3x · dxi ⊗ dxi

(δγ k d

dt

(ρg−1 d

dtg)

+ δγ k∂k12ρ|u|2)

.

Now we have the following proposition.

Proposition 4 Using the above def initions, the variational principle in SDiff(�),

δSfree[γ ]δγ (x, t)

∣∣∣SDiff(�)|t

= 0,

is reduced to the Euler equation,

∂

∂tρui + u j∂ jρui + ∂i p = 0, (3.6)

where p comes from the projection from TDiff(�)|SDiff(�) → TSDiff(�).

Proof Basically we leave the rigorous proof and especially the derivation of pto [5, 16]. The existence of p was investigated well in Appendix of [16] as theHodge decomposition [1, 36]. (See also the following Remark 1.) Except thederivation of p, we use the above relations and the following relations,

ddt

(ρe−γ j∂ j

ddt

eγ n∂n

)= d

dt

(ρui(γ (t), t)∂i

)

=(

∂

∂tρui|x=γ +

(ddt

γ j)

∂ jρui)

∂i

=(

∂

∂tρui + u j∂ jρui

)∂i

=:(

DDt

ρui)

∂i.

Then we obtain the Euler equation. �

Remark 1 The Euler equation was obtained by the simple variational prin-ciple. Physically speaking, the conservation of the momentum in the senseof Noether’s theorem [11, 24] led to the Euler equation. However, we couldintroduce the pressure pL term as the Lagrange multiplier of the constraint ofthe volume preserving. In the case, instead of Sfree, we deal with

Sfree,p = Sfree +∫

Tdt∫

�

pL(x, t)∂γ

∂xd3x.


Then noting the term coming from the Jacobian, the relation,

δSfree,p[γ ]δγ (x, t)

∣∣∣SDiff(�)|t

= 0,

is reduced to the Euler equation,

∂

∂tρui + u j∂ jρui + ∂i

(pL + 1

2ρ|u|2)

= 0.

As the pressure is determined by the (divergence free) condition of u, werenormalize [29, (25)],

p := pL + 12ρ|u|2.

More rigorous arguments are left to [16, 38] and physically interpretations are,e.g., in [2, 10, 26, 37, 40, 46].

We give a comment on the projection from TDiff(�)|SDiff(�) → TSDiff(�)

in Eq. (3.6), which is known as the projection method. First we note that thedivergence free condition div(u) = 0 simplifies the Euler equation (3.6),

ρDuDt

+ ∇ p = 0,∂ui

∂t+ u j∂ jui + 1

ρ∂i p = 0.

As mention in Section 6, in the difference equation we have a naturalinterpretation of the projection method [13]. We, thus, regard Du/Dt inTDiff(�)|SDiff(�) as lim

δt→0

u(t+δt)−u(t)δt for u(t + δt) := u(t + δt, γ (t + δt)) ∈ diff(�)

and u(t) := u(t, γ (t)) ∈ sdiff(�), i.e., div (u(t)) = 0 by considering TDiff(�) atthe unit e of Diff(�) up to δx2, as we did in Eqs. (3.3) and (3.4). In order tofind the deformation u‖(t + δt) in sdiff(�) by a natural projection from diff(�)

to sdiff(�) [14, ,p.36], we decompose u(t + δt) into u‖(t + δt) and u⊥(t + δt)such that ∂iu⊥i(t + δt) := ∂iui(t + δt). Then u‖(t + δt) := u(t + δt) − u⊥(t + δt)belongs to sdiff(�). Thus the pressure p is determined by [14]

∂iui(t + δt) + δt∂i1ρ

∂i p = 0. (3.7)

In other words, since u‖(t + δt) ≡ ui(t + δt) + δt 1ρ∂i p belongs to sdiff(�), the

deformation of u‖i(t + δt) − ui(t) which gives Du‖/Dt and the Euler equation(3.6) is the deformation in IFluid(� × T). After taking the continuous limitδt → 0, the equation for the pressure (3.7) can be written as [13],

(∂iu j)(∂ jui) + ∂i1ρ

∂i p = 0,

by noting the relations [∂t, ∂i] = 0 and div(u(t)) = 0, i.e., ∂iui(t + δt) =∂i[ui(t) + ∂

∂t ui(t)δt + u j(t)∂ jui(t)δt] + O(δt2). The Poisson equation with (3.6)

guarantees the divergence free condition. Hence the pressure p in the incom-pressible fluid is determined geometrically.


4 Reformulation of Surface Tension as a Minimal Surface Energy

In this section we reformulate the SURFER scheme [30] following the varia-tional principle and the arguments of previous sections.

4.1 Analytic Expression of Surface Area

We first should note that in general, the higher dimensional generalizedfunction like the Dirac delta function has some difficulties in its definition[47]. For the difficulties, in the Sato hyperfunctions theory [27], the sheaftheory and the cohomology theory are necessary to the descriptions of thehigher dimensional generalized functions, which are too abstract to be appliedto a problem with an arbitrary geometrical setting. Even for the generalizedfunction in the framework of Schwartz distribution theory, we should payattentions on its treatment. However since the surface S in this article is ahypersurface and its codimension is one, the situation makes the problemsmuch easier.

We assume that the smooth surface S is orientable and compact such thatwe could define its inner side and outer side. In other words, there is a threedimensional subspace (a manifold with boundary) B such that its boundary∂ B agrees with S and B is equal to the inner side of S with S itself. Then weconsider a generalized function θ over � ⊂ E

3 such that it vanishes over thecomplement Bc = � \ B and is unity for the interior B◦ := B \ ∂ B; θ is knownas a characteristic function of B.

We consider the global function θ(x) and its derivative dθ(x) in the sense ofthe generalized function, which is given by

dθ(x) =∑

i

∂iθ(x)dxi = ∂qθ(x)dq.

Here we use the notations in Section 2.1. Using the nabla symbol ∇θ =(∂iθ(x))i=1,2,3, |∇θ |d3x is interpreted as

|∇θ |d3x = |(∗dθ) ∧ dq|.Here due to the Hodge star operation ∗ : �p(�) → �3−p(�), ∗dθ = e∂qθds1 ∧ds2 where e is the Jacobian between the coordinate systems (ds1, ds2, dq) and(dx1, dx2, dx3). Then we have the following proposition;

Proposition 5 If the integral,

A :=∫

�

|∇θ |d3x ≡∫

�

|(∗dθ) ∧ dq|,

is f inite, A agrees with the area of the surface S.

It should be noted that due to the codimension of S ⊂ �, we have used thefact that the Dirac δ function along q ∈ TS is the integrable function whoseintegral is the Heaviside function. This fact is a key of this approach.


4.2 Quasi-characteristic Function for Surface Area

For the later convenience, we introduce a support of a function over �, whichis denoted by “supp”, i.e., for a function g over �, its support is defined by

supp(g) = {x ∈ � | g(x) �= 0},where “ ¯ ” means the closure as the topological space �.

One of our purposes is to express the surface S by means of numerical meth-ods, approximately. Since it is difficult to deal with the generalized functionθ in a discrete system like the structure lattice [15], we introduce a smoothfunction ξ over � as a quasi-characteristic function which approximates thefunction θ [9, 30],

ξ(x) =

⎧⎪⎪⎨

⎪⎪⎩

0 for x ∈ Bc⋂{x ∈ � | |q(x)| < εξ/2}c,

1 for x ∈ B⋂{x ∈ � | |q(x)| < εξ/2}c,

monotonicallyincreasing in q(x)

otherwise.

(4.1)

We note that along the line of dq for q ∈ (−εξ /2, εξ /2), ξ is a monotonicallyincreasing function which interpolates between 0 and 1. We now implicitlyassume that εξ is much smaller than εT defined in Section 2.1 so that supportof |∇ξ | is in the tubular neighborhood TS. However after formulating thetheory, we extend the geometrical setting in Section 2.1 to more general oneswhich include singularities; there εT might lose its mathematical meaning butεξ survives as a control parameter which governs the system. For example, asin [30], we can also deal with a topology change well.

By letting ξ c(x) := 1 − ξ(x), ξ c and ξ are regarded as the partition of unity[28, I p.272], or

ξ(x) + ξ c(x) ≡ 1.

We call these ξ and ξc “color functions” or “phase fields” in the followingsections. We have an approximation of the area of the surface S by thefollowing proposition.

Proposition 6 Depending upon εξ , we def ine the integral,

Aξ :=∫

�

|∇ξ |d3x,

and then the following inequality holds,

|Aξ − A| < εξ · A.

Here we note that Aξ is regarded as the approximation of the area A of Scontrolled by εξ . In other words, we use εξ as the parameter which controls thedifference between the characteristic function θ and the quasi-characteristicfunction ξ in the phase field model [9, 30].

Let us consider its extremal point following the variational principle in apurely geometrical sense.


Proposition 7 For sufficiently small εξ , we have

δ

δξ(x)Aξ = −∂i

∂iξ

|∇ξ | (x)

= κ(x),

where x ∈ S or q = 0.

Proof Noting the facts that ∂ξ/∂q < 0 at q = 0 and

|∇ξ | = √∇ξ · ∇ξ,

Proposition 2 and the equality in Proposition 1 show the relation. �

In the vicinity of S, q in Section 2.1 could be identified with the level-set function and the authors in [48, 49] also used this relation. Since allof geometrical quantities on S are lifted to TS as the inverse image of π ,the relation in Proposition 7 is also defined over (supp(|∇ξ |))◦ ⊂ TS and weredefine the κ by the relation from here.

4.3 Statics

Let us consider physical problems as we finish the geometrical setting. Beforewe consider dynamics of the phase field flow, we consider a statical surfaceproblem. Let σ be the surface tension coefficient between two fluids corre-sponding to ξ and ξc. Now let us call ξ and ξc “color functions” or “phasefields”. More precisely, we say that a color function with individual physicalparameters is a phase field. The surface energy E := σA is, then, approximatelygiven by

Etwo := σAξ = σ

∫

�

|∇ξ |d3x. (4.2)

As a statical mechanical problem, we consider the variational method of thissystem following Section 2.2.

Since a statical surface phenomenon is caused by the difference of thepressure of each material, we now consider a free energy functional [32],

Ftwo :=∫

�

(σ |∇ξ | − (p1ξ + p2ξ

c))

d3x, (4.3)

where pa (a = 1, 2) is the proper pressure of each material.

Proposition 8 The variational problem with respect to ξ , δFtwo/δξ = 0, repro-duces the Laplace equation [31, Chap.7],

(p1 − p2) − σκ(x) = 0, x ∈ (supp(|∇ξ |))◦. (4.4)

Proof As in Proposition 2, direct computations give the relation. �


This proposition implies that the functional Ftwo is natural. The solutionsof Eq. (4.4) are given by the constant mean curvature surfaces studied in[19, 20, 45].

Furthermore we also have another static equation, whose relation to theLaplace equation (4.4) is written in Remark 4.

Proposition 9 For every point x ∈ �, the variation principle, δFtwo/δxi = 0,gives

σ

⎛

⎝∑

j

∂i∂ jξ∂ jξ

|∇ξ | −∑

j

∂ j∂ jξ∂iξ

|∇ξ |

⎞

⎠− (p1 − p2)∂iξ = 0, (4.5)

or

∂ jτij(x) − (p1 − p2)∂iξ(x) = 0, (4.6)

where

τ(x) := σ

(I − ∇ξ

|∇ξ | ⊗ ∇ξ

|∇ξ |)

|∇ξ |(x).

Proof We are, now, concerned with the variation x → x + δx for every pointx ∈ �. We apply Proposition 3 to this case, i.e.,

δFtwo

δxi= −σ

[∂i|∇ξ | − ∂ j

(∂iξ(x) · δ

δ∂ jξ(x)|∇ξ |)]

(x) + (p1 − p2)∂iξ(x),

by using Eq. (4.4) as its Euler–Lagrange equation (2.3). Further for x �∈(supp(|∇ξ |))◦, its Euler–Lagrange equation (2.3) gives a trivial relation, i.e.,“0 = 0”. Then we have Eq. (4.6). �

Remark 2 It is worthwhile noting that Eqs. (4.5) and (4.6) are defined over �

rather than (supp(|∇ξ |))◦ because due to the relation,

|∂iξ | � |∇ξ |,even at the point at which denominators in the first term in Eq. (4.5) vanish,the first term is well-defined and vanishes.

Hence Eqs. (4.5) and (4.6) could be regarded as an extension of the definedregion of Eq. (4.4) to � and thus Eqs. (4.5) and (4.6) have the advantageover Eq. (4.4). The extension makes the handling of the surface tension mucheasier.

Remark 3 In the statical mechanics, there appears a force ∂iτij, which agreeswith one in (33) and (34) in [30] and (2.11) in [25]. We should note that in [25],it was also stated that this term is derived from the momentum conservationhowever there was not its derivation in detail. The derivation of the above τ

needs the Euler–Lagrange equation (2.3), which corresponds to the Laplace


equation (4.4) in this case, when we apply Proposition 3 to this system, thoughthese objects did not appear in [25].

Remark 4 In this remark, we comment on the identity between Eqs. (4.4) and(4.6). Comparing these, we have the identity,

∂iτij = σ∂ jξ · κ,

which is, of course, obtained from the primitive computations. It implies thatEq. (4.6) can be derived from the Laplace equation (4.4) with this relation.However it is worthwhile noting that both come from the variational prin-ciple in this article. In fact, when we handle multiple junctions, we need ageneralization of the Laplace equations over there like Eq. (5.7), which is noteasily obtained taking the primitive approach. Further the derivations from thevariational principle show their geometrical meaning in the sense of [1, 4, 11].

4.4 Dynamics

Now we investigate the dynamics of the two-phase field. There are twodifferent liquids which are expressed by phase fields ξ and ξ c respectively. Weassume that they obey the incompressible fluid dynamics. As in the previoussection, we consider the action of the volume-preserving diffeomorphismgroup SDiff(�) on the color functions ξ and ξ c. We extend the domain of ξ andξc to � × T and they are smooth sections of Coor(� × T). For the given t, wewill regard ξ and ξ c as functions of γ i in the previous section, i.e., ξ = ξ(γ (x, t)).For example, the density of the fluid is expressed by the relation,

ρ = ρ1ξc + ρ2ξ

for constant proper densities ρ1 and ρ2 of the individual liquids. The density ρ,now, differs from a constant function over � × T in general.

We consider the action integral Stwo including the surface energy,

Stwo[γ ] =∫

Tdt∫

�

(12ρ|u|2 − σ |∇ξ | + (p1ξ + p2ξ

c)

)d3x. (4.7)

The ratio between ρ and σ determines the ratio between the contributions ofthe kinematic part and the potential (or surface energy) part in the dynamicsof the fluid. Since the integrand in (4.7) contains no ∂ξ/∂t term, we obtain thesame terms in the variational calculations from the second and the third termin (4.7) as those in (4.4) and (4.6) in the static case even if we regard n as 4and x4 as t in Section 2.2. By applying Proposition 2 to this system, we have thefollowing proposition as the Euler–Lagrange equation for ξ .

Lemma 1 The function derivative of Stwo with respect to ξ gives

12(ρ1 − ρ2)|u(x, t)|2 + (p1 − p2) − σκ(x, t) = 0, x ∈ (supp(|∇ξ |))◦, (4.8)

up to the volume preserving condition.


This could be interpreted as a generalization of the Laplace equation (4.4)as in the following remark.

Remark 5 Here we give some comments on the generalized Laplace equation(4.8) up to the volume preserving condition. This relation (4.8) does not lookinvariant for Galileo’s transformation, u → u + u0 for a constant velocity u0.However for the simplest problem of Galileo’s boost, i.e., static state on asystem with a constant velocity u0, (4.8) gives

12(ρ1 − ρ2)|u0|2 + (p1 − p2) − σκ(x, t) = 0, x ∈ (supp(|∇ξ |))◦, (4.9)

which might differ from the Laplace equation (4.4). However for the boost, weshould transform pa into

pa := pa + 12ρa|u0|2. (4.10)

Then the above equation of pa agrees with the static one (4.4). In other wordsEq. (4.10) makes our theory invariant for the Gaililio’s transformation.

For a more general case, we should regard pa as a function over � × Trather than a constant number due to the volume preserving condition. Thesevalues are contained in the pressure as mentioned in Eq. (4.12). The statement“up to the volume preserving condition” has the meaning in this sense. Infact, in the numerical computation, these individual pressures pa’s are not soimportant as we see in Remark 6. Due to the constraint of the incompressible(volume-preserving) condition, the pressure p is determined as mentioned inRemark 1. There are no contradictions with the Galileo’s transformation andSDiff(�)-action.

We consider the infinitesimal action of SDiff(�) around its identity. As didin Section 3, we apply the variational method to this system in order to obtainthe Euler equation with the surface tension.

Proposition 10 For every (x, t) ∈ � × T, the variational principle,δStwo/δγ

i(x, t) = 0, gives the equation of motion, or the Euler equationwith the surface tension,

Dρui

Dt+ σ

⎛

⎝∑

j

∂i∂ jξ∂ jξ

|∇ξ | −∑

j

∂ j∂ jξ∂iξ

|∇ξ |

⎞

⎠+ ∂i p = 0. (4.11)

Here p is also the pressure coming from the ef fect of the volume-preserving.

Proof The measure d3x is regarded as ∂γ

∂x d3x with ∂γ

∂x = 1. Noting ddt

∂γ

∂x = 0, theproof in Proposition 4 and Remark 1 provide the kinematic part with pressureterm and Proposition 9 gives the remainder. In this proof, the total pressure pis defined in Remark 6. �


Remark 6 More rigorous speaking, as we did in Remark 1, we also renormalizethe pressure

p = pL + 12ρ|u|2 + p1ξ + p2ξ

c

= pL + 12(ρ1 − ρ2)ξ |u|2 + (p1 − p2)ξ + 1

2ρ2|u|2 + p2. (4.12)

As in Section 2.2, the third term in (4.11) includes the effects from pa’s via thegeneralized Laplace equation (4.8) as the Euler–Lagrange equation (2.3).

Remark 7

1. The equation of motion (4.11) is the same as (24) in [30] basically. Weemphasize that it is reproduced by the variational principle.

2. As in [30], in our framework, we can deal with the topology changes andthe singularities which are controlled by the parameter εξ . The abovedynamics is well-defined as a field equation provided that εξ is finite. Ifneeds, one can evaluate its extrapolation for vanishing of εξ .

3. In general, εξ is not constant for the time development. Due to theequation of motion, it changes. At least, in numerical computation, thenumerical diffusion makes the intermediate region wider in general. How-ever even when the time passes but we regard it as a small parameter, theapproximation is justified.

4. Since from Remark 2, the surface tension is defined over �, the Eulerequation is defined over � without any assumptions.

5. It should be noted that the surface force is not difficult to be computed asin [30] but there sometimes appear so-called parasite current problems inthe computations even though we will not touch the problem in this article.

5 Multi-phase Flow with Multiple Junctions

In this section, we extend the SURFER scheme [30] of two-phase flow to multi-phase (N-phase, N � 2) flow.

5.1 Geometry of Color Functions

In order to extend the geometry of the color functions in the previous section,we introduce several geometrical tools. First let us define a geometrical objectsimilar to smooth d-manifold with boundary. Here we note that d-manifoldmeans d-dimensional manifold, and d-manifold with boundary means that itsinterior is a d-manifold and its boundary is a (d − 1)-dimensional manifold. Wedistinguish a smooth (differential) manifold from a topological manifold here.

When we consider multi-junctions in E3, we encounter a geometrical object

with smooth “boundaries” whose dimensions are two, one and zero eventhough it is regarded as a topological 3-manifold with boundary.


Definition 1 We say that a path-connected topological d-manifold with bound-ary V is a path-connected interior smooth d-manifold if V satisfies thefollowing:

1. The interior V◦ is a path-connected smooth d-manifold, and2. V has finite path-connected subspaces Vα , (α = 1, · · · , ) such that

(a) V \ V◦ is decomposed by Vα , i.e.,

V \ V◦ = ∐

α=1

Vα,

(b) Each Vα is a path-connected smooth k-manifold in � (k < d).

We say that Vα is a singular-boundary of V and let their union V \ V◦ denotedby ∂singV := V \ V◦.

Here the disjoint union is denoted by∐

, i.e., for subsets A and B of �,A∐

B := A⋃

B if A⋂

B = ∅.By letting V(n) := V and V[k] := {Vα ⊂ V | dim Vα � k}, and by picking up

an appropriate path-connected part V(k) ⊂ V[k] each k, we can find a naturalstratified structure,

V(n) ⊃ V(n−1) ⊃ · · · ⊃ V(2) ⊃ V(1) ⊃ V(0),

which is known as a stratified submanifold in the singularity theory [6].In terms of path-connected interior smooth d-manifolds, we express subre-

gions corresponding to materials in a regions � ⊂ E3 as extensions of B and

Bc in Section 4.1.

Definition 2 For a smooth domain � ⊂ E3, we say that N path-connected

interior smooth 3-manifolds {Ba}a=0,··· ,N−1 are colored decomposition of � if{Ba}a=0,··· ,N−1 satisfy the following:

1. every Ba is a closed subset in �,2. � =⋃a=0,··· ,N−1 Ba, and3. � \ (

⋃a<b Ba ∩ Bb ) =∐a=0,··· ,N−1 B◦

a.

Roughly speaking, each Ba corresponds to a material in �; Definition 2 (1)means that Ba is surrounded by singular boundary or the boundary of �, (2)implies that there is no “vacuum” in � and (3) guarantees that the interiors ofthese materials don’t overlap.

In general, for a �= b , Ba ∩ Bb is a singular geometrical object if it is notthe empty set. Singularity basically makes its treatment difficult in math-ematics. In order to avoid such difficulties, we introduce color functionsξa(x) (a = 0, 1, 2, · · · , N − 1) over a region �, which are modeled on ξ and ξc

as in Section 4.1, are controlled by a small parameter εξ > 0 and approximatethe characteristic functions over Ba.


To define color functions ξa(x) (a = 0, 1, 2, · · · , N − 1), we introduce an-other geometrical object, ε-tubular neighborhood in E

3:

Definition 3 For a closed subspace U ⊂ � and a positive number ε, ε-tubularneighborhood TU,ε of U is defined by

TU,ε :={

x ∈ � | dist(x, U) <ε

2

},

where dist(x, U) is the distance between x and U in E3.

We assume that each T∂sing Ba,ε has a fiber structure over ∂sing Ba as topologicalmanifolds as mentioned in Section 2.1. Using the ε-tubular neighborhood, wedefine εξ -controlled color functions.

Definition 4 We say that N smooth non-negative functions {ξa}a=0,··· ,N−1 over� ⊂ E

3 are εξ -controlled color functions associated with a colored decomposi-tion {Ba}a=0,··· ,N−1 of �, if they satisfy the following:

1. ξa belongs to C∞(�) and for x ∈ �,∑

a=0,1··· ,N−1

ξa(x) ≡ 1.

2. For every Ma := supp(ξa) and La := supp(1 − ξa), (a = 0, 1, · · · , N − 1),

(a) Ba � Ma,(b) Lc

a � Ba,(c) (Ma \ Lc

a)◦ ⊂ T∂sing Ba,εξ

,(d) (Ma \ Lc

a)◦ ⊃ ∂sing Ba.

3. For x ∈ (Ma \ Lca), we define the smooth function qa by

qa(x) ={

dist(x, ∂sing Ba), x ∈ (Ma \ Ba),

−dist(x, ∂sing Ba), otherwise.

Then for the flow exp(−t ∂∂qa

) on C∞(�)|(Ma\Lca)

, ξa monotonically increasesalong t ∈ U ⊂ R at x ∈ (Ma \ Lc

a).

When (Ma \ Lca)

◦ = T∂sing Ba,εξfor every a, {ξa}a=0,··· ,N−1 are called proper εξ -

controlled color functions associated with the colored decomposition of � ⊂ E3,

{Ba}a=0,··· ,N−1 or merely proper.

The functions ξa’s are, geometrically, the partition of unity [28, I p.272] anda quasi-characteristic function, roughly speaking, which is equal to 1 in thefar inner side of Ba, vanishes at the far outer side of Ba and monotonicallybehaves in the artificial intermediate region. Noting that the flow exp(−t ∂

∂q )

corresponds to the flow from the outer side to the inner side, ξa decreases fromthe inner side to the outer side.


From here, let us go on to use the notations Ba, Ma, La, and ξa in Definition4. Further for later convenience, we employ the following assumptions whichare not essential in our theory but make the arguments simpler.

Assumption 1 We assume the following:

1. The colored decomposition {Ba}a=0,··· ,N−1 of � and εξ satisfy the conditionthat every Lc

a is not the empty set.This assumption means that the singularities that we consider can beresolved by the above procedure. Since εξ can be small enough, thisassumption does not have crucial effects on our theory.

2. The colored decomposition {Ba}a=0,··· ,N−1 of � and εξ satisfy the relation,

∂�⋂⎛

⎝⋃

a �=b ;a,b �=0

Ma

⋂Mb

⎞

⎠ = ∅,

and every intersection Ba⋂

B0 perpendicularly intersects with ∂�.This describes the asymptotic behavior of the materials. For example M0will be assigned to a wall in Section 6. This assumption is neither soessential in this model but makes the arguments easy of the boundaryeffect. As mentioned in Section 3, we neglect the boundary effect becausewe are concerned only with a local theory or differential equations. If onewishes to remove this assumption, he could consider smaller region �′ ⊂ �

after formulates the problems in �.3. The volume of every Ba, the area of every ∂sing Ba, and the length def ined

over every one-dimensional object in ∂sing Ba are f inite.As our theory is basically local, this assumption is not essential, either.

Under the assumptions, we fix colored decomposition {Ba}a=0,··· ,N−1 and εξ -controlled color functions {ξa}a=0,··· ,N−1.

As mentioned in the previous section, we have an approximate descriptionof the area of ∂sing Ba.

Proposition 11 By letting the area of ∂sing Ba be Aa, the integral

Aξa :=∫

�

|∇ξa|d3x,

approximates Aa by

|Aξa − Aa| < εξAa.

Here we notice that Mab := Ma⋂

Mb (a, b = 0, 1, 2, · · · , N − 1, a �= b)

means the intermediate region whose interior is a 3-manifold. Similarlywe define Mabc := Ma

⋂Mb⋂

Mc (a, b , c = 0, 1, 2, · · · , N − 1; a �= b , c; b �=c) and so on. Since the relation,

⋃Ma = �, holds, we look on the intersections

of Ma as an approximation of the intersections of Ba which is parameterizedby εξ . Even though there exist some singular geometrical objects in {Ba}


[6], we can avoid its difficulties due to finiteness of εξ . (We expect that thecomputational result of a physical process might have weak dependence onεξ which is small enough. More precisely the actual value is obtained bythe extrapolation of εξ = 0 for series results of different εξ ’s approaching toεξ = 0.)

5.2 Surface Energy

Let us define the surface energy E (N)exact by

E (N)exact =

∑

a>b

σabArea(

Ba

⋂Bb

),

where σab is the surface tension coefficient (σab > 0, σab = σba) between thematerials corresponding to Ba and Bb , and Area(U) is the area of an interiorsmooth 2-manifold U .

We have an approximation of the surface energy E (N)exact by the following

proposition.

Proposition 12 The free energy,

E (N) =∑

a>b

σab

∫

�

d3x√|∇ξa(x)||∇ξb (x)|(ξa(x) + ξb (x)), (5.1)

has a positive number M such that

|E (N) − E (N)exact| < εξ M.

Proof For a �= b , Ba⋂

Bb consists of the union of some interior smooth 2-manifolds. Their singular-boundary parts ∂sing(Ba

⋂Bb ) ≡ {Vα}α∈Iab are union

of some smooth 1-manifolds and smooth 0-manifolds. Thus {Vα}α∈Iab has noeffect on the surface energy E (N)

exact because they are measureless.Over the subspace,

Mpropab := {x ∈ Mab | ξa(x) + ξb (x) = 1}◦, (5.2)

and for a positive number , we have identities,

|∇ξa(x)|(ξa(x) + ξb (x)) = |∇ξb (x)|(ξa(x) + ξb (x))

= √|∇ξa(x)||∇ξb (x)|(ξa(x) + ξb (x)) .(5.3)

The sum of the integrals over Mpropab dominates E (N) if εξ is sufficiently small.

We evaluate the remainder. For example, for different a, b and c, the partin E (N) coming from

Mpropabc := {x ∈ Mabc | ξa(x) + ξb (x) + ξc(x) = 1}◦ (5.4)


is order of εξ2. Namely we have

∣∣∣∣

∫

Mabc

d3x√|∇ξa(x)||∇ξb (x)|(ξa(x) + ξb (x)) − Length(Ba ∩ Bb ∩ Bc)

∣∣∣∣

< εξ2Length(Ba ∩ Bb ∩ Bc),

where Length(C) is the length of a curve C. Thus we find a number Msatisfying the inequality. �

Remark 8

1. M is bound by

M � max(σab)

×(∑

a<b

(Area(Ba ∩ Bb ) + εξ Length

(∂sing(Ba ∩ Bb )

))+ Kεξ2

)

,

where K is the number of isolated points in all of singular-boundary partsof {Ba}.

2. It should be noted that E (N) becomes the surface energy of the systemexactly when εξ vanishes.

3. Using the identities (5.3), we can also approximate E (N) by

∑

a �=b

σab

∫

�

d3x |∇ξa(x)|(ξa(x) + ξb (x)) ,

using a positive number . In such a way, there are so many variants which,approximately, represent the surface energy in terms of ξa’s.

5.3 Statics

Let us consider the statics of the multi-phase fields. In the above argumentsin this section, we have given the geometrical objects, first, and defined thefunctions ξa, functional energy E (N) and so on. In this subsection on thestatic mechanics of the multi-fields, we consider the deformation of thesegeometrical objects and determine a configuration whose corresponds to anextremal point of the functional, i.e., Fmul in the following proposition. In otherwords, we derive the Euler–Lagrange equation which governs the extremalpoint of the functional and characterizes the configuration of Ma, La andapproximately Ba for every a = 0, · · · , N − 1.

Let us introduce the proper pressure

pP(x) :=∑

paξa(x), (5.5)

where pa is a certain pressure of each material.


Proposition 13 The Euler–Lagrange equation of the static free energy integral,

Fmul =∫

�

⎛

⎝∑

a �=b

σab

√|∇ξa(x)||∇ξb (x)|(ξa(x) + ξb (x)) − pP

⎞

⎠d3x,

with respect to ξa, i.e., δFmul/δξa = 0, is given as follows:

1. For a point x ∈ Mabprop of (5.2),

(pa − pb ) − κa(x) = 0, (5.6)

where

κa := −∂i∂iξa

|∇ξa| .

2. For a point x ∈ Mpropabc of (5.4),

(pa − pb − pc) − κabc(x) = 0, (5.7)

where

κabc := √|∇ξb (x)||∇ξc(x)| −√|∇ξa(x)||∇ξb (x)| −√|∇ξa(x)||∇ξc(x)|

+ ∂i

[∂iξa√|∇ξa|3

.(√|∇ξb |(ξa + ξb ) +√|∇ξc|(ξa + ξc)

)]

. (5.8)

Proof For a point x ∈ Mabprop of (5.2), we have ξa(x) + ξb (x) = 1, and thus the

Euler–Lagrange equation (2.3) leads (5.6).Similarly for a point x ∈ Mabc

prop of (5.4), we have ξa(x) + ξb (x) + ξc(x) = 1,and thus the concerned terms of the integrand in the energy functional aregiven by

· · · +√|∇ξa(x)||∇ξb (x)|(ξa + ξb ) +√|∇ξa(x)||∇ξc(x)|(ξa + ξb )

+√|∇ξb (x)||∇ξc(x)|(1 − ξa) + · · · . (5.9)

The Euler–Lagrange equation (2.3) gives (5.7). �

Remark 9

1. It is noticed that (5.6) agrees with the Laplace equation (4.4) and thus wealso reproduce the Laplace equation locally.

2. Equation 5.7 could be regarded as another generalization of the Laplaceequation though Mprop

abc does not contribute to the surface energy when εξ

vanishes and has a negligible effect even for a finite εξ if εξ is sufficientlysmall. Indeed, (5.7) does not appear in the theory of surface tension [31].However (5.7) is necessary and plays a role to guarantee the stability in the


numerical computations and to preserve the consistency in of numericalapproach with finite intermediate regions for εξ �= 0.

3. Similarly we have similar equations for a higher intersection regions.

As a generalization of Eq. (4.5) we immediately have the following.

Proposition 14 For every point x ∈ �, the variational principle, δFmul/δxi = 0,gives

∂i pP−∑

a �=b

σa,b

[

∂i

(√|∇ξa||∇ξb |(ξa + ξb ))−∂ j

(∂iξa∂ jξa√|∇ξa|3

√|∇ξb |(ξa + ξb )

)]

=0.

(5.10)

Proof It is the same as Proposition 9, which essentially comes fromProposition 3. �

Remark 10 In Proposition 14, we can apply the equation without any clas-sification of geometry like Eqs. (5.2) and (5.4). It is also noted that Eq. (5.10)is globally defined over � as mentioned in Remark 2.

5.4 Dynamics

Using these equations, let us consider the dynamics of the multi-phase flow.We extend the colored-decomposition of � and the εξ -controlled color func-tions of {ξa}a=0,··· ,N−1 to those of � × T and C∞(� × T) using another fiberstructure of Coor(� × T). Mathematically speaking, since our space-time is atrivial bundle � × T and has the fiber structure � × (ta, tb ) → � for a smallinterval (ta, tb ) due to the integrability, we can consider the pull-back of themap ξa : � → R. If we consider a global behavior of ξa with respect to timet, we should pay more attentions on the Lagrange picture γ (x, t) and theintegrability. However as our theory is local, we can regard (ta, tb ) as T with aninfinitesimal interval.

Thus ξa is redefined as ξa := ξa(γ (x, t)) for (x, t) ∈ � × T and it is denoted byξa(x, t). In the time development of ξa, the control parameter εξ is not necessaryto be constant. However in this article, we assume that εξ is sufficiently smallfor every t ∈ T.

Let the density of each ξa be denoted by ρa. We have the global densityfunction ρ(x, t) and pressure pP(x, t) given by

ρ(x, t) =∑

ρaξa(x, t), pP(x, t) =∑

paξa(x, t).


In contrast to the previous subsection, in this subsection, we investigatean initial problem. In other words, every configuration of the geometricalobjects, Ma, La and approximately Ba (a = 0, · · · , N − 1), with divergencefree velocity u, (div(u) = 0) can be an initial condition to the dynamics ofthe multi-phase fields. The following equations which we will derive in thissubsection govern the deformations of these geometrical objects as their time-development. Further it is noticed that in this subsection, the proper pressurepP(x, t) has no mathematical nor physical meaning because it becomes a partof the total pressure p, which is determined by the divergence free conditiondiv(u) = 0 as mentioned in Remark 1.

We have the first theorem;

Theorem 1 The action integral of the multi-phase f ields, or the εξ -controlledcolor functions ξa with physical parameters ρa, σab, pa (a, b = 0, 1, · · · , N − 1)

def ined above, is given by

Smul =∫

Tdt∫

�

⎛

⎝12ρ|u|2 −

∑

a �=b

σab

√|∇ξa||∇ξb |(ξa + ξb ) + pP

⎞

⎠d3x, (5.11)

under the volume-preserving deformation.

Proof The action integral is additive. The first term exhibits the kinematicenergy of the fluids. The second term represents the surface energy up to εξ

as in Proposition 12. The proper pressure pP in Eq. (5.5) leads the Laplaceequations. We can regard it as the action integral of the multi-phase fields withthese parameters. �

Then we have further generalization of Eq. (4.8) as follows:

Lemma 2 Assume that every Ma(t), Mpropab (t) and Mprop

abc (t) deform for the time-development following a certain equation. The Euler–Lagrange equation ofthe action integral with respect to ξa, δSmul/δξa = 0, is given, up to the volumepreserving condition, as follows:

1. For a point x ∈ Mpropab , we have

12(ρa − ρb )|u(x, t)|2 + (pa − pb ) − κa(x, t) = 0. (5.12)

2. For a point x ∈ Mpropabc , we have

12(ρa − ρb − ρc)|u(x, t)|2 + (pa − pb − pc) − κabc(x, t) = 0. (5.13)

Similarly we have the similar equations for higher intersection regions.

Proof It is the same as proof of Proposition 13. �


Using these equations, we have the second theorem, which is our maintheorem:

Theorem 2 For every (x, t) ∈ � × T, the variational principle, δSmul/δγ (x, t) =0, provides the equation of motion,

Dρui

Dt+ ∂i p +

∑

a �=b

σa,b

[

∂i

(√|∇ξa||∇ξb |(ξa + ξb ))

− ∂ j

(∂iξa(x)∂ jξa(x)

√|∇ξa|3√|∇ξb |(ξa + ξb )

)]

= 0. (5.14)

Here p is the pressure coming from the ef fect of the volume-preserving orincompressible condition, which includes the proper pressure pP (5.5).

Proof We naturally obtain it by using (1) Proposition 4 and its proof, (2)Remark 1, (3) Lemma 2 and (4) Proposition 3. �

Here we note that by expressing the low-dimensional geometry in termsof the global smooth functions ξ ’s with finite εξ , we have unified the infinitedimensional geometry or the incompressible fluid dynamics governed byIFluid(� × T), and the εξ -parameterized low dimensional geometry with sin-gularities to obtain the extended Euler equation (5.14). When εξ approachesto zero, we must consider the hyperfunctions [23, 27] instead of C∞(� × T),but we conjecture that our results would be justified even under the limit; theunification would have more rigorous meanings.

It should be noted that on the unification, it is very crucial that we expressthe low-dimensional geometry in terms of the global smooth functions ξ ’s asthe infinite-dimensional vector spaces. The SDiff(�) naturally acts on ξ ’s andthe thus we could treat the low-dimensional geometry and the incompress-ible fluid dynamics in the framework of the infinite dimensional Lie group[5, 16, 38]. It is contrast to the level-set method. As mentioned in Section 2.1,the level-set function does not belong to C∞(�) and thus we can not considerSDiff(�) action and treat it in the framework.

Remark 11

1. Equation 5.14 is the Euler equation with the surface tension of multi-phasefields which gives the equation of motion for the multi-phase flow evenwith the multiple junctions. As we will illustrate examples in Section 6, thedynamics with the triple junction can be solved without any geometricalconstraints. It should also noted that for a point in Mprop

ab , Eq. (5.14) isreduced to the original Euler equation in [30] or Eq. (4.11).

2. The Euler equation (5.14) appears as the momentum conservation in thesense of Noether’s theorem (Section 2.2). It implies that Eq. (5.14) isnatural from the geometrical viewpoint [4, 5, 16, 26, 29, 33, 37].


3. Further even though we set {ξa(·, t)} as proper εξ -controlled coloredfunctions as an initial state, their time-development is not guaranteedthat {ξa(·, t)}, (t > 0), is proper εξ -controlled. In general εξ may becomelarge for the time development, at least, numerically due to the numericaldiffusion. (See examples in Section 6). However even for t > 0, we can findεξ (t) such that {ξa(·, t)} are εξ (t)-controlled colored functions and if εξ (t) issufficiently small, our approximation is guaranteed by εξ (t).

4. The surface tension is also defined over � × T and thus the Euler equationis defined over � × T without any assumptions due to Remark 2.

5. We may set εξ depending upon the individual intermediate region betweenthese fields by letting εab mean that for ξa and ξb , a �= b . Then if we

recognize εξ asN−1maxa,b=0

εab, above arguments are applicable for the case.

6. We defined the εξ -controlled colored functions using the εT -tubular neigh-borhood TU,εT and the colored decomposition of � in Definition 4 byletting εT = εξ . On the other hand, as in [30], our formulation can describea topology change well following the Euler equation (5.14) such as asplit of a bubble into two bubbles in a liquid. The εξ -controlled coloredfunctions can represents the geometry for such a topology change withoutany difficulties. However on the topology change, the path-connectedregion and the εξ -tubular neighborhood lose their mathematical meaningand thus, more rigorously, we should redefine the εξ -controlled coloredfunctions. Since the εξ -controlled colored functions represent the geometryas an analytic geometry, it is not difficult to modify the definitions though itis too abstract. In other words, we should first define the εξ -controlled col-ored functions ξ ’s without the base geometry, and characterize geometricalobjects using the functions ξ ’s. However since such a way is too abstract tofind these geometrical meanings, we avoided a needless confusion in thesedefinitions and employed Definition 4.

5.5 Equation of Motion of Triple-phase Flow

Let us concentrate ourselves on a triple-phase flow problem, noting Eq. (5.3).From the symmetry of the triple phase, we introduce “proper” surface tensioncoefficients,

σ0 = σ01 + σ02 − σ12

2, σ1 = σ01 + σ12 − σ02

2, σ2 = σ02 + σ12 − σ01

2,

or σab = σa + σb . Here it should be noted that the “proper” surface tensioncoefficient is based upon the speciality of the triple-phase and does not havemore physical meaning than above definition.


Lemma 3 For dif ferent a, b , and c, we have the following approximation,∣∣∣∣

∫

�

(√|∇ξa||∇ξb |(ξa + ξb ) +√|∇ξa||∇ξc|(ξa + ξc) − |∇ξa|)

d3x

∣∣∣∣ < εξAa.

(5.15)

Using the relation, the free energy Eq. (5.1) has a simpler expressionup to εξ .

Proposition 15 By letting

E (3)sym := σ0

∫

�

d3x |∇ξ0(x)| + σ1

∫

�

d3x |∇ξ1(x)| + σ2

∫

�

d3x |∇ξ2(x)|,

we have a certain number M related to area of the surfaces {Ba} such that

|E (3) − E (3)sym| < εξ M.

Proof Due to Lemma 3, it is obvious. �

The action integral Eq. (5.11) also becomes

Stri =∫

Tdt∫

�

(12ρ|u|2 −

∑

a

(σa|∇ξa| − paξa)

)

d3x.

For a practical reason, we consider a simpler expression by specifying theproblem.

5.6 Two-phase Flow and Wall with Triple-junction

More specially we consider the case that ξo corresponds to the wall which doesnot move. For the case, we can neglect the wall part of the equation, becauseit causes a mere energy-shift of E (3)

sym. Then the action integral and the Eulerequation become simpler. We have the following theorem as a corollary.

Theorem 3 The action integral of two-phase f low with wall is given by

Swall =∫

Tdt∫

�

(12ρ|u|2 −

2∑

a=1

(σa|∇ξa| − paξa)

)

d3x,

and the equation of motion is given by

Dρui

Dt+ ∂i p − ∂ j(τ ij) = 0, (5.16)

where

τ =2∑

a=1

σa

(I − ∇ξa

|∇ξa| ⊗ ∇ξa

|∇ξa|)

|∇ξa|. (5.17)


Practically this Euler equation (5.16) is more convenient due to the propersurface tension coefficients. However this quite differs from the original (4.5)and (4.6) in [30] and governs the motion of two-phase flow with a wallcompletely.

Remark 12 Equation 5.16 is the Euler equation with the surface tension oftwo-phase fields with a wall or triple junctions in our theoretical framework.We should note that under the approximation (5.15), (5.16) is equivalent toEq. (5.14), even though Eq. (5.16) is far simpler than Eq. (5.14).

From Remark 2, it should be noted that τ and the Euler equation (5.16) aredefined over � × T. This property as a governing equation is very importantfor the computations to be stable, which is mentioned in Introduction. Sincethe non-trivial part of τ is localized in � of each t ∈ T, τ vanishes and has noeffect on the equation in the other area.

We will show some numerical computational results of this case in thefollowing section. There we could also consider the viscous stress forces andthe wall shear stress.

6 Numerical Computations

In this section, we show some numerical computations of two-phase flowsurrounded by a wall obeying the extended Euler equation in Theorem 3.As in Theorem 3, the wall is expressed by the color function ξ0 and has theintermediate region (M0 \ Lc

0)◦ where ξ0 has its value (0, 1). As dynamics of

the incompressible two-phase flow with a static wall, we numerically solve theequations,

div(u) = 0,

Dρui

Dt+ (∂i p − Ki) = 0,

Dρ

Dt= 0. (6.1)

Here for the numerical computations, we assume that the force K consists ofthe surface tension, the viscous stress forces, and the wall shear stress,

K j = ∂iτij + ∂iτij + τ j. (6.2)

Here τ is given by Eq. (5.17), τ is the viscous tensor,

τij := 2η

(Eij − 1

3div(u)

), Eij := 1

2

(∂ui

∂x j+ ∂u j

∂xi

)

with the viscous constant

η(x) := η1ξ1 + η2ξ2,


and τ j is the wall shear stress which is localized at the intermediate region(M0 \ Lc

0)◦ where ξ0 has its value (0, 1).

The boundary condition of the interface between the fluid ξa (a = 1, 2) andthe wall ξ0 is generated dynamically in this case. In other words, in orderthat the wall shear stress term suppress the slip over the intermediate region(M0 \ Lc

0)◦ asymptotically t → ∞ due to damping, we let τ j be proportional to

j-component of ∂u‖/∂q0 for the parallel velocity u‖ to the wall and relevantto (1 − ξ0(x)), and make u vanish over L0. Here q0, M0, and L0 are ofDefinition 4.

The viscous force can not be dealt with in the framework of the Hamiltoniansystem because it has dissipation. However from the conventional consider-ation of the balance of the momentum [16, Section 13], it is not difficult toevaluate it. The viscosity basically makes the numerical computations stable.

In the numerical computations, we consider the problem in the structurelattice L marked by aZ

3, where Z is the set of the integers and a is a positivenumber. The lattice consists of cells and faces of each cell. Let every cell be acube with sides of the length a. We deal with a subspace �L of the lattice as�L := � ∩ L ⊂ E

3. The fields ξ ’s are defined over the cells as cellwise constantfunctions and the velocity field u is defined over faces as facewise constantfunctions [15]; ξ is a constant function in each cell and depends on the positionof the cell, and similarly the components of the velocity field, u1, u2, and u3 arefacewise constant functions defined over x2x3-faces, x3x1-faces, and x1x2-facesof each cell respectively.

As we gave a comment in Remark 11(5), we make the parameter εξ dependon the intermediate region in this section. Let ε12 be the parameter for thetwo-phase field or the liquids, and ε0 := ε01 ≡ ε02 be one for the intermediateregion (M0 \ Lc

0)◦ between liquids and the wall.

As mentioned in Introduction, we assume that ε12 for the two-phase fieldin our method is given as ε12 � a so that we could estimate the intermediateeffect in our model following [7, 9, 12, 25, 30], even though the thickness ofthe intermediate region among real liquids is of atomic order and is basicallynegligible in the macroscopic theory.

In the computational fluid dynamics, the VOF (volume of fluid) methoddiscovered by Hirt and his coauthors [15, 21, 22] is well-established when wedeal with fluid with a wall. Since we handled triple-junction problems as inSection 5.6, we reformulate our model in the VOF-method. It implies that weidentify 1 − ξ0 with the so-called V-function V := 1 − ξ0 in the VOF methodbecause V in the VOF method means the volume fraction of the fluid andcorresponds to 1 − ξ0 in our formulation.

As the convention in [21], V is also defined as a cellwise constant function. Inthe following examples, we will set ε0 to be a or the unit cell basically. Howeverwe can also make it ε0 > a as for two-phase field. It means that for the caseε0 > a, we consider each cell as a fictitious porous material whose volume ratioV ∈ [0, 1] without imposing any wall shear stress on the fictitious surface ofthe porous parts itself in each cell as in Fig. 1. (As mentioned above, we setthe wall shear stress τ j from the physical wall ξ0. The porous parts are purely


Fig. 1 VOF with porousmatter expression: for theconsistency between the colorfunction and VOF-method,we consider each cell as afictitious porous materialwhose volume ratio and openfraction are a value in [0, 1]without imposing any wallshear stress on fictitioussurface of the porous parts ineach cell. This expressionrepresents purely geometricaleffects

V

position

1

0

fictitious.) The region where V is equal to 1 means the region where fluid freelyexists whereas the region where V vanishes means the region where existenceof fluid is prohibited. The region with V ∈ (0, 1) is the intermediate region(M0 \ Lc

0)◦. Here we emphasize that the fictitious porous in each cell brings

purely geometrical effects to this model.Then we could go on to consider the problem in consistency between VOF-

method and ξ0 function in the phase-field model. Let functions f1 ≡ f and f2over supp(V) be defined by the relations,

ξ1 = V f1, ξ2 = V f2, f1 + f2 = 1.

Further we also modify the open fraction A in the VOF-method, which isdefined over each face. We interpret A as the open area of the fictitious porousmaterial of each face of each cell, which also has a value in [0, 1] as in Fig. 1.We also use the open area fraction A of each face of each cell [21, 22]. Fora face belonging to the cell whose V = 1, A is also equal to 1. Following theconvention in discretization by Hirt [21], A is regarde valued functions like

A ◦ u ≡ Au = (A1u1, A2u2, A3u3) ,

(Au)1 = A1u1, (Au)2 = A2u2, (Au)3 = A3u3. (6.3)

Here we note that Aia2 implicitly appearing in (6.3) can be interpreted as atwo-chain of homological base associated with a face of a cell. For example,for a velocity field μ := ui(x)dxi defined over a cell in the continuous theoryand a piece of the boundary element of the cell A1a2, the discretized u1 definedover the face is given by

(Au)1 := 1a2

∫

A1a2∗μ = A1u1,

where ∗ is the Hodge star operator, i.e., ∗μ := u1(x)dx2dx3 + u2(x)dx3dx1 +u3(x)dx1dx2. Thus the discretization (6.3) is very natural even from the pointof view of the modern differential geometry.


Hence div(u) ≡ ∇u reads ∇ Au as the difference equation in VOF-method[21] and we employ this discretization method.

We give our algorithm to compute (6.1) precisely as follows. As a con-vention, we specify the quantities with “old” and “new” corresponding tothe previous states and the next states at each time step respectively in thecomputation. In other words, we give the algorithm that we construct the nextstates using the previous data by regarding the current state as an intermediatestate in the time step. We use the project-method [13, 15];

I : ρu − ρuold

�t= − (uold · ∇) ρuold,

II : unew − u�t

= − 1ρ

(∇ p − K),

III : ∇unew = 0.

The step I is the part of the advection of the velocity uold. In the step I, wedefine an intermediate velocity u and after then, we compute unew and p in thesteps II and III.

The time-development of ρ is given by the equation,

f new = f old + �t∇ ((Auold) f old) ,

and

ρ = V(ρ1 f + ρ2(1 − f ))

for the proper densities ρa of ξa (a = 1, 2).Even for the case that we can deal with multi-phase flow with large density

difference, we evaluate its time-development. Precisely speaking, when weevaluate u, following the idea of Rudman [39] we employ the momentumadvection u of u,

u := 1ρnew

[ρolduold − �t

(uold · ∇) ρolduold] .

Our derivation of the Euler equation shows that the Rudman’s method is quitenatural.

Following the conventional notation, the guessed-value of the velocity isdenoted by u∗, which is the initial value for the steps in II and III. Let us define

u∗ := u + �t1

ρnewK(ρold, f old, uold) .

In order to evaluate the guessed velocity, we compute the force K from (6.2)noting that divτ and divτ read ∇ Aτ and ∇ Aτ respectively.

Following the SMAC (Simplified-Marker-and-Cell) method [3, 13, 15], wenumerically determine the new velocity unew and the pressure p in a certain


boundary condition using the preconditioned conjugate gradient method(PCGM):

(IIa) Evaluate p using the PCGM : 1�t

∇(A ◦ u∗) = ∇ A ◦ 1ρnew

∇ p,

(IIb) By using p determine unew : unew = u∗ − �t1

ρnew∇ p.

More precisely speaking, (III) ∇(A ◦ unew) = 0 means that we numericallysolve the Poisson equation,

∇(

A�t1

ρnew∇ p)

= ∇(A ◦ u∗).

Then we obtain unew, which obviously satisfies (III) ∇(A ◦ unew) = 0, whichis known as the Hodge decomposition method [3, 13, 14] as mentioned inRemark 1.

Following the algorithm, we computed the two-phase flow with a wall andtriple junctions. We illustrate two examples of the numerical solutions of thetriple junction problems as follows.

6.1 Example 1

Here we show a computation of a capillary problem, or the meniscus os-cillation, in Fig. 2. We set two liquids in a parallel wall with the physicalparameters; η1 = η2 = 0.1 [cp], ρ1 = ρ2 = 1.0 [pg/μm3], σ1 = 3.349 [pg/μsec2],σ2 = 46.651 [pg/μsec2].

Fig. 2 The meniscusoscillation: each figure showsthe time development

t=1.0[µsec] t=2.0[µsec]t=0.0[µsec]

t=4.0[µsec] t=5.0[µsec]t=3.0[µsec]

30[degree]


We used L := 12 [μm] × 0.5 [μm] × 16 [μm] lattice whose unit length a is0.125 [μm]. The first liquid exists in the down side and the second liquid doesin the upper side in the region 10 [μm] × 0.5 [μm] × 15 [μm] surrounded bythe wall and the boundaries with the boundary conditions. As the boundaryconditions, at the upper side from the bottom of the wall by 15 [μm], we fixthe constant pressure as 100 [KPa] and, along x2-direction, we set the periodicboundary condition.

We set ε12 = ε0 = 1 mesh for the intermediate regions, at least, as its initialcondition. Each time interval is 0.001 [μsec].

As the initial state, we start the state that the fluid surface is flat as in Fig. 2aand the first liquid exists in the box region 10 [μm] × 0.5 [μm] × 7.0 [μm],which is not stable. Due to the surface tension, it moves and starts to oscillatebut due to viscosity, the oscillation decays. Though we did not impose thecontact angle as a geometrical constraint, the dynamics of the contact angle wascalculated due to a balance between the kinematic energy and the potentialenergy or the surface energy. The oscillation converged to the stable shapewith the proper contact angle, which is given by

cos ϕ = σ2 − σ1

σ2 + σ1≡ σ02 − σ01

σ12. (6.4)

The angle given by σ ’s are designed as 30 [degree] whereas it in the numericalexperiment in Fig. 2 is a little bit larger than 30 [degree], though it is verydifficult to determine it precisely. However since we could tune the parametersσ ’s so that we obtain the required state, our formulation is very practical.

Due to the numerical diffusions and others, the thickness of the intermedi-ate regions changes in the time development and also depends on the positionsof the interfaces, even though it is fixed the same at the initial state. Howeverwe consider that it is thin enough to evaluate the physical system since thecontact angle is reasonably estimated.

6.2 Example 2

This example is on the computations of the contact angles for different surfacetension coefficients displayed in Fig. 3.

Even in this case, in order to see the difference between the designed contactangle and computed one, we go on to handle two-dimensional symmetricalproblems though we used three-dimensional computational software. In otherwords, we set that x2-direction is periodic.

Since the contact angle ϕ in our convention is given by the formula (6.4). Bysetting σ ’s

σ1

σ2= 1 − cos ϕ

1 + cos ϕ,

for given the contact angle ϕ, we computed five triple junction problemswithout any geometrical constraints; each σ is given in the caption in Fig. 3.


(a)

(b)

(c)

(d)

(e)

30[degree]

45[degree]

60[degree]

90[degree]

120[degree]

Fig. 3 The different contact angles are illustrated due to the different surface energy:by σ1 = 1.0000 [pg/μsec3], a ϕ = 30 [degree], σ2 = 13.9282 [pg/μsec3], b ϕ = 45 [degree],σ2 = 5.8284[pg/μsec3], c ϕ = 60 [degree], σ2 = 3.0000 [pg/μsec3], d ϕ = 90 [degree], σ2 =1.0000 [pg/μsec3], and e ϕ = 120 [degree], σ2 = 0.3333 [pg/μsec3]

The other physical parameters are given by η1 = η2 = 0.1 [cp] and ρ1 = ρ2 =1.0 [pg/μm3].

In this computation we used a 240 × 4 × 112 lattice whose unit length a is0.125 [μm]; � = 30 [μm] × 0.5 [μm] × 14 [μm]. We set the flat layer as a wallby thickness 3 [μm] from the bottom of � along the z-axis. As the boundaryconditions, at the upper side from the bottom of the wall by 9 [μm], we fix theconstant pressure as 100 [KPa].

As the initial state for each computation. we set a semicylinder with radius5 [μm] in the flat wall like Fig. 3d. We also set ε12 = ε0 = 1 mesh for theintermediate regions. Each time step also corresponds to 0.001 [sec].

Due to the viscosity, after time passes sufficiently 50 [μsec], the staticsolutions were obtained as illustrated in Fig. 3, which recover the contact anglesunder our approximation within good agreements.

7 Summary

By exploring an incompressible fluid with a phase-field geometrically [4, 5,16, 26, 29, 33, 37], we reformulated the expression of the surface tension forthe two-phase flow found by Lafaurie et al. [30] as a variational problem.We reproduced the Euler equation of two-phase flow (4.11) following thevariational principle of the action integral (4.7) in Proposition 10.

The new formulation along the line of the variational principle enabled usto extend Eq. (4.11) to that for the multi-phase (N-phase, N � 2) flow. By


extending Eq. (4.11), we obtained the novel Euler equation (5.14) with thesurface tension of the multi-phase fields in Theorem 2 from the action integralof Theorem 1 as the conservation of momentum in the sense of Noether’s the-orem. The variational principle for the infinite dimensional system in the senseof [4, 5, 16] gives the equation of motion of multi-phase flow controlled by thesmall parameter εξ without any geometrical constraints and any difficulties forthe singularities at multiple junctions.

For the static case, we gave governing equations (5.6), (5.7) and (5.10) whichgenerate the locally constant mean curvature surfaces with triple junctions bycontrolling a parameter εξ to avoid these singularities. As the solutions of Eq.(4.4) has been studied well as the constant mean curvature surfaces for lasttwo decades [18–20, 45], our extended equations (5.6), (5.7) and (5.10) mightshed new light on treatment of singularities of their extended surfaces, or aset of locally constant mean curvature surfaces. (Even though we need aninterpretation of our scheme, for example, it can be applied to a soap filmproblem with triple junction.) It implies that our method might give a methodof resolutions of singularities in the framework of analytic geometry.

By specifying the problem of the multi-phase flow to the contact angleproblems at triple junctions with a static wall, we obtained the simpler Eulerequation (5.16) in Theorem 3. Using the VOF method [21, 22], we showed twoexamples of the numerical computations in Section 6. In our computationalmethod, for given surface tension coefficients, the contact angle is automati-cally generated by the surface tension without any geometrical constraints andany difficulties for the singularities at triple junctions. The computations werevery stable. It means that the computations did not collapse nor behave wildlyfor every initial and the boundary conditions.

In our theoretical framework, we have unified the infinite dimensional geo-metry or an incompressible fluid dynamics governed by IFluid(� × T), andthe εξ -parameterized low dimensional geometry with singularities given bythe multi-phase fields. We obtained all of equations following the samevariational principle. We naturally reproduced the Laplace equations, (4.4)and (5.6), and obtained their generalizations (4.8), (5.6), (5.7), (5.13) and(5.10), and the Euler equations, (4.11), (5.14), and (5.16) in Proposition 10 andTheorems 2 and 3. These equations are derived from the same action integralsby choosing the physical parameters. In the sense of [1, 4, 11], it implies thatwe gave geometrical interpretations of the multi-phase flow. Even thoughthe phase-field model has the artificial intermediate regions with unphysicalthickness εξ , our theory supplies a model which shows how to evaluate theireffects on the surface tension forces, from geometrical viewpoints. The key factof the model is that we express the low-dimensional geometry in terms of theinfinite-dimensional vector spaces, or global functions ξ ’s which have naturalDiff and SDiff actions. Thus we can treat them in the framework of infinitedimensional Lie group [5, 16, 38] to consider its Euler equation. It is contrastto the level-set method; in analytic geometry and algebraic geometry, zerosof a function expresses a geometrical object and thus the level-set method isso natural from the point of view. However as mentioned in Section 2.1, the


level-set function cannot be a global functions as C∞(�) and thus it is difficultto handle the method in the framework of the infinite dimensional Lie groupSDiff(�).

As our approach gives a resolution of the singularities by a parameter εξ , infuture we will explore topology changes, geometrical objects with singularitiesand so on, more concretely in our theoretical framework. When εξ approachesto zero, we need more rigorous arguments in terms of hyperfunctions [27] butwe conjecture that our results would be correct for the vanishing limit of εξ

because the Heaviside function is expressed by θ(q) = limεξ →0

1π

tan−1(

qεξ

)in the

Sato hyperfunction theory, which could be basically identified with ξ(q) ofthe finite εξ . Since an application of the Sato hyperfunction theory to fluiddynamics was reported by Imai on vortex layer and so on [23], we believe thatthis approach might give another collaboration between pure mathematics andfluid mechanics.

Acknowledgements This article is written by the authors in memory of their colleague, collab-orator and leader Dr. Akira Asai who led to develop this project. The authors are also gratefulto Mr. Katsuhiro Watanabe for critical discussions and to the anonymous referee for helpful andcrucial comments.

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Math Phys Anal Geom (2011) 14:279–294DOI 10.1007/s11040-011-9097-z

Weak Convergence and Banach Space-ValuedFunctions: Improving the Stability Theoryof Feynman’s Operational Calculi

Lance Nielsen

Received: 23 December 2010 / Accepted: 4 July 2011 / Published online: 13 August 2011© Springer Science+Business Media B.V. 2011

Abstract In this paper we investigate the relation between weak convergenceof a sequence {μn} of probability measures on a Polish space S convergingweakly to the probability measure μ and continuous, norm-bounded functionsinto a Banach space X. We show that, given a norm-bounded continuousfunction f : S → X, it follows that limn→∞

∫S f dμn = ∫

S f dμ—the limit onehas for bounded and continuous real (or complex)—valued functions on S.This result is then applied to the stability theory of Feynman’s operationalcalculus where it is shown that the theory can be significantly improved overprevious results.

Keywords Feynman’s operational calculus · Disentangling · Stability theory ·Weak convergence of probability measures

Mathematics Subject Classifications (2010) Primary 60B10 · 60B12;Secondary 46B28 · 47A56 · 81S99

1 Introduction

The primary area of investigation of this paper is the establishment of arelation between Banach space valued functions defined on a metric space andweak convergence of sequences of probability measures on the metric space.In particular, suppose we have a metric space S and a weakly convergent se-quence {μn}∞n=1 of probability measures on S such that μn ⇀ μ, μ a probability

L. Nielsen (B)Department of Mathematics, Creighton University, Omaha, NE 68178, USAe-mail: [email protected]

280 L. Nielsen

measure on S. Given a Banach space X and a continuous and norm boundedfunction f : S → X, is it true that

limn→∞

∫

Sf dμn =

∫

Sf dμ? (1)

If X is a separable Hilbert space, the answer was shown to be “yes” in thepaper [15], also by the current author. Given an arbitrary Banach space, onemay not expect that the answer is “yes”; however, in Theorem 2 it is shownthat one can indeed establish (1) for norm-bounded continuous functions intoa Banach space X.

In the last section of the paper, we address the use of the theorem provedin Section 2 to improve the stability theory for Feynman’s operational calculus[Johnson et al., in preparation, 12, 16–19]. Before going further, we provide ashort discussion of Feynman’s operational calculus.

Feynman’s operational calculus originated with the 1951 paper [4] andconcerns itself with the formation of functions of non-commuting operators.Indeed, even functions as simple as f (x, y) = xy are not well-defined if xand y do not commute. Indeed, some possibilities are f (x, y) = yx, f (x, y) =12 (xy + yx), and f (x, y) = 1

3 xy + 23 yx. One then has to decide, usually with a

particular problem in mind, how to form a given function of non-commutingoperators. One method of dealing with this problem is the approach developedby Jefferies and Johnson in the series of papers [5–8] and expanded on in thepapers [9, 12], and others. The Jefferies–Johnson approach to the operationalcalculus uses measures on intervals [0, T] to determine the order of operatorsin products. In the original setting used by Jefferies and Johnson, only contin-uous measures were used. However, Johnson and the current author extendedthe operational calculus to measures with both continuous and discrete partsin [12].

The discussion above, then, begs the question of how measures can be usedto determine the order of operators in products. Feynman’s heuristic rules forthe formation of functions of non-commuting operators give us a starting point.

(1) Attach time indices to the operators to specify the order of operators inproducts.

(2) With time indices attached, form functions of these operators by treatingthem as though they were commuting.

(3) Finally, “disentangle” the resulting expressions; i.e. restore the conven-tional ordering of the operators.

As is well known, the central problem of the operational calculus is thedisentangling process. Indeed, in his 1951 paper, [4], Feynman points out that“The process is not always easy to perform and, in fact, is the central problemof this operator calculus.”

We first address rule (1) above. It is in the use of this rule that we will seemeasures used to track the action of operators in products. First, it may bethat the operators involved may come with time indices naturally attached.For example, we might have operators of multiplication by time dependent

Weak Convergence and Vector-Valued Functions 281

potentials. However, it is also commonly the case that the operators used areindependent of time. Given such an operator A, we can (as Feynman mostoften did) attach time indices according to Lebesgue measure as follows:

A = 1t

∫ t

0A(s) ds

where A(s) := A for 0 � s � t. This device does appear a bit artificial butdoes turn out to be extremely useful in many situations. We also note thatmathematical or physical considerations may dictate that one use a measuredifferent from Lebesgue measure. For example, if μ is a probability measureon the interval [0, T], and if A is a linear operator, we can write

A =∫

A(s) μ(ds)

where once again A(s) := A for 0 � s � T. When we write A in this fashion,we are able to use the time variable to keep track of when the operator Aacts. Indeed, if we have two operators A and B, consider the product A(s)B(t)(here, time indices have been attached). If t < s, then we have A(s)B(t) = ABsince here we want B to act first (on the right). If, on the other hand, s < t,then A(s)B(t) = BA since A has the earlier time index. In other words, theoperator with the smaller (or earlier) time index, acts to the right of (or before)an operator with a larger (or later) time index. (It needs to be kept in mind thatthese equalities are heuristic in nature.) For a much more detailed discussionof using measures to attach time indices, see Chapter 14 of the book [10] andalso Chapter 2 of the forthcoming book (Johnson et al., in preparation) andthe references contained in both.

Concerning the rules (2) and (3) above, we mention that, once we haveattached time indices to the operators involved, we calculate functions of thenon-commuting operators as if they actually do commute. These calculationsare, of course, heuristic in nature but the idea is that with time indicesattached, one carries out the necessary calculations giving no thought to theoperator ordering problem; the time indices will enable us to restore thedesired ordering of the operators once the calculations are finished; this isthe disentangling process and is typically the most difficult part of any givenproblem.

We now move on to discuss, in general terms, how the operational calculuscan be made mathematically rigorous. Suppose that Ai : [0, T] → L(X), i =1, . . . , n, are given and that we associate to each Ai(·) a Borel probabilitymeasure μi on [0, T]; this is the so-called time-ordering measure and, asmentioned above, serves to keep track of when a given operator or operator-valued function acts in products. We construct a commutative Banach algebra(the disentangling algebra) DT

((A1(·), μ1)

∼ , . . . , (An(·), μn)∼)

of functions an-alytic on a certain polydisk. With this commutative Banach algebra in hand, wecan carry out the disentangling calculations called for by Feynman’s “rules” ina mathematically rigorous fashion. Once the disentangling is carried out in thealgebra DT , we map the result to the non-commutative setting of L(X) using

282 L. Nielsen

the so-called disentangling map T Tμ1,...,μn

; it is the image under the disentanglingmap that is the disentangled operator given by the application of Feynman’s“rules”. We note that changing the n–tuple of time-ordering measures will, ingeneral, change the operational calculus, as it will usually change the action ofthe disentangling map. Of course, a change in the operators will also generallychange the operational calculus. A much more detailed discussion of this ap-proach to the operational calculus can be found in Johnson et al., in preparation.

The stability theory for the Jefferies–Johnson formulation of the opera-tional calculus was developed initially in [19] and expanded on in [11, 15, 17,18], and [16]. In particular, stability with respect to the time-ordering measures,the focus of the last section of the current paper, can be described as follows.We select sequences {μik}∞k=1 of Borel probability measures on [0, T] such thatμik ⇀ μi as k → ∞. We then have, for each k ∈ N, a particular operationalcalculus, given by the action of T T

μ1k,...,μnk, indexed by the n–tuple (μ1k, . . . , μnk)

of measures and thus a sequence of operational calculi. The stability question isthen the question of whether the sequence of operational calculi has a limitingoperational calculus as k → ∞.

As an example of such a stability theorem, we state the theorem to whichTheorem 2 will be applied. This theorem is Theorem 3.1 of [18].

Theorem 1 Let Ai : [0, T] → L(X), i = 1, . . . , n, be continuous with respect tothe usual topology on [0, T] and the norm topology on L(X). Associate toeach Ai(·) a continuous Borel probability measure μi on [0, T]. Let {μik}∞k=1,i = 1, . . . , n, be sequences of Borel probability measures on [0, T] such that, foreach i = 1, . . . , n, μik ⇀ μi. Construct the direct sum Banach algebra

UD :=∑

k∈N∪{0}

⊕DT

((A1(·), μ1k)

∼ , . . . , (An(·), μnk)∼)

where for k = 0 the summand is DT((A1(·), μ1)

∼ , . . . , (An(·), μn)∼)

. Then

limk→∞

∣∣�

(T T

μ1k,...,μnk(πk(θ f ))

) − �(T T

μ1,...,μn(π0(θ f ))

)∣∣ = 0

for all � ∈ L(X)∗ and all θ f = ( f, f, f, . . .) ∈ UD.

The conclusion of this theorem requires the selection of an element � ∈L(X)∗ in order to obtain a bounded and continuous real or complex valuedfunction to which to apply the usual ideas of weak convergence of measures.The improvement gained by establishing Theorem 2 and allows us to eliminatethe need to select an element of the dual of L(X) and instead allows us tosimply select a vector φ ∈ X; it will then follow that

limk→∞

∥∥T T

μ1k,...,μnk( f )φ − T T

μ1,...,μn( f )φ

∥∥

X= 0;

that is, we have strong operator convergence, a much improved situation overthe theorem above.

As remarked below, in the discussion that begins Section 3, Theorem 2 canbe applied to many of the theorems contained in [18] and [16] to improve the


stability theorems contained in these papers. Our consideration of the theoremabove allows us to illustrate a typical application of Theorem 2.

2 The Main Theorem

For our main theorem on the connection between weak convergence ofprobability measures and Banach space valued functions, we will take X tobe a Banach space. Further, we will take S to be a Polish space. (That is, S isa separable topological space that admits a complete metric. See [1, p.73], forexample.)

Remark 1 In the application of Theorem 2 below to Feynman’s operationalcalculus, we will take S to be [0, T]m for a positive integer m. Of course, [0, T]m

is a Polish space and so Theorem 2 can be applied.

As we will be working with weakly convergent sequences of probabilitymeasures on S, we recall the definition of weak convergence. (See, for exam-ple, [2].)

Definition 1 Let {μn}∞n=1, μ, be Borel probability measures on the metric spaceS. We say that the sequence {μn}∞n=1 converges weakly to μ as n → ∞ if

limn→∞

∫

Sg dμn =

∫

Sg dμ (2)

for every bounded, continuous, real-valued function g on S. We will denoteweak convergence by μn ⇀ μ.

Remark 2 In this paper we will use continuous and bounded complex-valuedfunctions. It is easily seen that, for such functions, (2) remains true. Indeed,for any bounded and continuous f : S → C, we can write f (s) = u(s) + iv(s),where u(s) and v(s) are real-valued continuous and bounded functions on S.Since

∫

Sf (s) dμn(s) =

∫

Su(s) dμn(s) + i

∫

Sv(s) dμn(s),

we see that it is still the case that (2) holds.

We will also take a moment to remind the reader of the definition of a tightfamily of probability measures. (See [2], page 37.)

Definition 2 A family � of probability measures on a metric space S is tightif for every positive ε there is a compact set K ⊆ S such that μ(K) > 1 − ε forevery μ ∈ �.

With these brief preliminaries out of the way, we can state the theorem.

284 L. Nielsen

Theorem 2 Let X be a Banach space. Let S be a Polish space and supposethat {μk}∞k=1 and μ are Borel probability measures on S such that μk ⇀ μ. Letf : S → X be continuous with respect to the norm topology on X and the metrictopology on S. Assume that

sups∈S

‖ f (s)‖X < ∞. (3)

Then

limk→∞

∫

Sf dμk =

∫

Sf dμ (4)

in norm on X.

Proof Let ‖ f‖ := sups∈S ‖ f‖X and choose ε > 0. Since {μk}∞k=1 is weaklycompact, {μk}∞k=1 is a tight family of measures, by Prohorov’s theorem (seeTheorems 6.1 and 6.2 of [2]). Hence there is a compact Kε ⊆ S such that

μk (S\Kε) <ε

4‖ f‖ (5)

for all k ∈ N. It then follows that

μ (S\Kε) <ε

4‖ f‖ (6)

as well. Define, for each k ∈ N,

νk := μk − μ. (7)

Using the Hahn-Banach theorem (see, for example, Corollary 1.6.2 of [13]),there is a sequence

{x∗

k

}∞k=1 in the unit ball of X∗ for which∥∥∥∥

∫

Kε

f dνk

∥∥∥∥

X

=∣∣∣∣

∫

Kε

x∗k ( f ) dνk

∣∣∣∣ (8)

for each k = 1, 2, . . .. Next, extract a subsequence{μkl

}∞l=1 of {μk}∞k=1 such that

lim supk→∞

∥∥∥∥

∫

Sf dνk

∥∥∥∥

X= lim

l→∞

∥∥∥∥

∫

Sf dνkl

∥∥∥∥

X. (9)

Since Kε ⊆ S is compact, f (Kε) is compact and so is a separable subset ofX. There is, by the Banach-Alaoglu theorem (see Corollary 1.6.5(i) of [13]),

a subsequence y∗n := x∗

klnof

{x∗

kl

}∞l=1

and a y∗ in the unit ball of X∗ with the

property that

limn→∞ y∗

n( f (s)) = y∗( f (s)) (10)

or, equivalently,

limn→∞(y∗

n − y∗)( f (s)) = 0 (11)


for s ∈ S. Define gn : S → C by gn(s) := (y∗n − y∗)( f (s)). The sequence

{gn(·)}∞n=1 is uniformly bounded since

∣∣y∗

n( f (s)) − y∗( f (s))∣∣ �

∣∣y∗

n( f (s))∣∣ + ∣

∣y∗( f (s))∣∣ � 2‖ f‖. (12)

Further, the sequence {gn(·)}∞n=1 is equicontinuous on Kε by virtue of thecontinuity of f . To see this, let s0 ∈ Kε . By the continuity of f there is aneighborhood U of s0 such that, for s ∈ U , ‖ f (s) − f (s0)‖X < ε/2. Then, usingthe definition of gn,

|gn(s) − gn(s0)| = ∣∣(y∗

n − y∗)( f (s) − f (s0))∣∣ � 2‖ f (s) − f (s0)‖X < ε, (13)

for any n ∈ N. Next, it’s clear that the set {gn(·)}∞n=1 is closed (as a subsetof C(S, C)). Hence we can apply the Arzela-Ascoli theorem [14, pages 277,279] to obtain a subsequence

{gnr

}∞r=1 and a g ∈ C (Kε, C) such that gnr → g

uniformly on Kε . But the pointwise limit of{gnr

}∞r=1 is the zero function and

so, for all s ∈ Kε , g(s) = 0. To simplify the notation in what follows, we will letσn := μkln

and recall that y∗n = x∗

kln. We may write

∥∥∥∥

∫

Sf dσn −

∫

Sf dμ

∥∥∥∥

X�

∥∥∥∥

∫

Kε

f d(σn − μ)

∥∥∥∥

X

+∥∥∥∥

∫

S\Kε

f d(σn − μ)

∥∥∥∥

X

�∣∣∣∣y

∗n

(∫

Kε

f (s) d(σn − μ)(s))∣

∣∣∣

+ ‖ f‖ (σn (S\Kε) + μ (S\Kε))

�∣∣∣∣

∫

Kε

y∗n ( f (s)) d(σn − μ)(s)

∣∣∣∣

+ ‖ f‖ (σn (S\Kε) + μ (S\Kε))

�∣∣∣∣

∫

Kε

(y∗

n − y)( f (s)) d (σn − μ) (s)

∣∣∣∣

+∣∣∣∣

∫

Kε

y∗ ( f (s)) d (σn − μ) (s)

∣∣∣∣ + ε

2(by (6))

� 2 sups∈Kε

∣∣(y∗

n − y)( f (s))

∣∣

+∣∣∣∣

∫

Kε

y∗ ( f (s)) d (σn − μ) (s)

∣∣∣∣ + ε

2

� 2 sups∈Kε

∣∣(y∗

n − y)( f (s))

∣∣

+∣∣∣∣

∫

Sy∗ ( f (s)) d (σn − μ) (s)

∣∣∣∣

286 L. Nielsen

+∣∣∣∣

∫

S\Kε

y∗ ( f (s)) d (σn − μ) (s)

∣∣∣∣ + ε

2

� 2 sups∈Kε

∣∣(y∗

n − y)( f (s))

∣∣

+∣∣∣∣

∫

Sy∗ ( f (s)) d (σn − μ) (s)

∣∣∣∣

+‖ f‖(

ε

4‖ f‖ + ε

4‖ f‖)

+ ε

2

= 2 sups∈Kε

∣∣(y∗

n − y)( f (s))

∣∣

+∣∣∣∣

∫

Sy∗ ( f (s)) d (σn − μ) (s)

∣∣∣∣ + ε. (14)

We know that σn ⇀ μ as n → ∞. Using the choice of the sequence{

y∗n

}∞n=1, it

follows from the last line above that

lim supk→∞

∥∥∥∥

∫

Sf dμk −

∫

Sf dμ

∥∥∥∥

X= lim

n→∞

∥∥∥∥

∫

Sf dσn −

∫

Sf dμ

∥∥∥∥

X� ε. (15)

Because ε was arbitrary,

lim supk→∞

∥∥∥∥

∫

Sf dμk −

∫

Sf dμ

∥∥∥∥

X= 0. (16)

This finishes the proof. �

3 Application to Feynman’s Operational Calculi

In this section we will address the application of the theorems above toFeynman’s operational calculi. In particular, the theorems above are mostdirectly applicable to the stability theory of the operational calculi in the time-dependent setting (see [18] and [15, 16]). Indeed, it is the work that the presentauthor has done in developing the stability theory for Feynman’s operationalcalculus that motivated the results contained in this paper and in [15]. Beforeaddressing the stability theory for the operational calculus, we will brieflysketch the relevant definitions for the time-dependent operational calculus.(See also [6–10, Johnson et al., in preparation], for example.)

The key object in our approach to the operational calculus is the disentan-gling map. In the time independent setting, this map and some of its importantproperties was developed by B. Jefferies and G. W. Johnson in the papers[5–8]. (The details of Jefferies’ and Johnson’s approach to the operationalcalculus and many extensions/applications of their approach can also be foundin the forth-coming book (Johnson et al., in preparation)). Before definingthe map, however, we need some preliminary definitions and notation (see[5, 9, 12]). (In fact, we follow the paper [9] as well as [5] even though the


later paper was concerned only with the time-dependent setting and we arehere concerned with the time-dependent setting.) We begin by introducing twocommutative Banach algebras AT and DT (the role of T will become apparentbelow). These algebras are closely related and play an important role in therigorous development of the operational calculus.

Given n ∈ N and n positive real numbers r1, . . . , rn, let AT(r1, . . . , rn) or,more briefly AT , be the space of complex-valued functions (z1, . . . , zn) �→f (z1, . . . , zn) of n complex variables that are analytic at the origin and are suchthat their power series expansion

f (z1, . . . , zn) =∞∑

m1,...,mn=0

am1,...,mn zm11 · · · zmn

n (17)

converges absolutely at least in the closed polydisk |z1| � r1, . . . , |zn| � rn. Allof these functions are analytic at least in the open polydisk |z1| < r1, . . . , |zn| <

rn. We remark that the entire functions of (z1, . . . , zn) are in AT(r1, . . . , rn) forany n–tuple (r1, . . . , rn) of positive real numbers.

For f ∈ AT given by equation (17) above, we let

‖ f‖ = ‖ f‖AT :=∞∑

m1,...,mn=0

|am1,...,,mn |rm11 · · · rmn

n . (18)

This expression is a norm on AT and turns AT into a commutative Banachalgebra. (See [5] or Johnson et al., in preparation for details. In fact, AT is aweighted �1-space.)

We now turn to the construction of the Banach algebra DT . To give the mostgeneral definition, we will let X be a Banach space and let Ai : [0, T] → L(X),i = 1, . . . , n, be measurable in the sense that A−1

i (E) is a Borel set in [0, T] forevery strongly open subset E of L(X). Associate to each Ai(·), i = 1, . . . , n, acontinuous Borel probability measure μi on [0, T] . We now define n positivereal numbers r1, . . . , rn by

ri :=∫

[0,T]‖Ai(s)‖L(X)dμi(s) (19)

for each i = 1, . . . , n. These real numbers will serve as weights and we ignorefor the present the nature of the Ai(·) as operator-valued functions andintroduce a commutative Banach algebra DT((A1(·), μ1)

∼, . . . , (An(·), μn)∼)

(the disentangling algebra) of “analytic functions” f ((A1(·), μ1)∼, . . . , (An(·),

μn)∼) or, more briefly written, f (A1(·)∼, . . . , An(·)∼) where the objects

(A1(·), μ1)∼ , . . . , (An(·), μn)

∼ or, more briefly, A1(·)∼, . . . , An(·)∼ replace theindeterminates z1, . . . , zn. For brevity, we will usually refer to the disentanglingalgebra as DT . (We write (Ai(·), μi)

∼ for the objects replacing z1, . . . , zn tostress that these objects depend not only on the operator-valued functions butalso on the measures we associate with them.) It is worth noting here thatthe operator-valued functions do not have to be distinct though we will stillconsider the formal objects obtained from them to be distinct in the Banach

288 L. Nielsen

algebra DT . Having said this, we take DT((A1(·), μ1)

∼ , . . . , (An(·, μn)∼)

to bethe collection of all expressions of the form

f (A1(·)∼, . . . , An(·)∼) =∞∑

m1,...,mn=0

am1,...,mn (A1(·)∼)m1 · · · (An(·)∼)

mn (20)

with the norm defined by

‖ f‖DT :=∞∑

m1,...,mn=0

∣∣am1,...,mn

∣∣ rm1

1 · · · rmnn . (21)

Via coordinate-wise addition and multiplication of such expressions it easilyfollows that equation (21) is a norm. Similarly, coordinate-wise addition andmultiplication of the expressions seen in (20) makes DT into a commutativeBanach algebra. (See Proposition 1.2 of [5].) Moreover, the Banach algebrasAT and DT can be identified. (See Proposition 1.3 of [5]. The proof ofProposition 1.3 in [5] is, of course, given in the time independent settingalthough it turns out that the proof in the time-dependent setting is the same.)

We work here in the commutative setting of the disentangling algebra DT .The definition of the disentangling map will depend on the disentangling ofthe monomial

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) := (A1(·)∼)m1 · · · (An(·)∼)

mn . (22)

Also, it is the disentangling of the monomial that shows best the connectionbetween Feynman’s ideas and this theory.

We now introduce the notation that is necessary for the disentangling map.For m ∈ N, let Sm be the set of all permutations of the integers {1, . . . , m} andgiven π ∈ Sm, we let

�m(π) := {(s1, . . . , sm) ∈ [0, T]m : 0 < sπ(1) < · · · < sπ(m) < T

}. (23)

When π is the identity permutation it is common to write �m(π) as �m.For j = 1, . . . , n and all s ∈ [0, T], we let

A j(s)∼ = A j(·)∼; (24)

that is, we discard the time dependence of the operator-valued functionsthough we will use the time index to keep track of when a given operator acts.Next, given nonnegative integers m1, . . . , mn and letting m = m1 + · · · + mn,we define

Ci(s)∼ :=

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

A1(s)∼ if i ∈ {1, . . . , m1},A2(s)∼ if i ∈ {m1 + 1, . . . , m1 + m2},

......

An(s)∼ if i ∈ {m1 + · · · + mn−1 + 1, . . . , m},(25)

for i = 1, . . . , m and s ∈ [0, T]. Even though Ci(s)∼ clearly depends on thenonnegative integers m1, . . . , mn, we will suppress this dependence in ournotation to ease the presentation.


We are now prepared to time-order the monomial Pm1,...,mn according to thedirections provided by the measures μ1, . . . , μn. We note that the calculationleading to the time-ordered expression below are much more complicated thanthe corresponding calculation found in Proposition 2.2 of [5] for continuoustime ordering measures. The details of the calculation can be found in [5, 9],and, in more detailed form, in Johnson et al., in preparation. We simply quotethe result here.

Proposition 1 Let m1, . . . , mn ∈ N be given. Then the monomial Pm1,...,mn(A1(·)∼,. . . , An(·)∼) is given in time ordered form by

Pm1,...,mn (A1(·)∼, . . . , An(·)∼)

=∑

π∈Sm

∫

�m(π)

Cπ(m)

(sπ(m)

)∼· · · Cπ(1)

(sπ(1)

)∼(μ

m11 × · · · × μmn

n

)(ds1, . . . , dsm) .

(26)

Now that we have the time-ordered monomial in hand, we can define thedisentangling map T T

μ1,...,μnwhich will take us from the commutative setting

of the disentangling algebra DT to the non-commutative setting of L(X). Allthat we need to do is replace the objects Ci(s)∼ by the corresponding operator-valued functions. This amounts to erasing the tildes; to be precise we define

Ci(s) :=

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

A1(s) if i ∈ {1, . . . , m1},A2(s) if i ∈ {m1 + 1, . . . , m1 + m2},

......

An(s) if i ∈ {m1 + · · · + mn−1 + 1, . . . , m}.(27)

Definition 3 We define the action of the disentangling map T Tμ1,...,μn

on themonomial Pm1,...,mn byT T

μ1,...,μnPm1,...,mn (A1(·)∼, . . . , An(·)∼)

:=∑

π∈Sm

∫

�m(π)

Cπ(m)

(sπ(m)

) · · · Cπ(1)

(sπ(1)

) (μ

m11 × · · · × μmn

n

)(ds1, . . . , dsm) .

(28)Then, for f (A1(·)∼, . . . , An(·)∼) ∈ DT (A1(·)∼, . . . , An(·)∼) given by

f (A1(·)∼, . . . , An(·)∼) =∞∑

m1,...,mn=0

am1,...,mn(A1(·)∼)m1 · · · (An(·)∼)mn , (29)

we set

T Tμ1,...,μn

f (A1(·)∼, . . . , An(·)∼)

:=∞∑

m1,...,mn=0

am1,...,mnT Tμ1,...,μn

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) . (30)

290 L. Nielsen

The disentangling map as defined here does indeed give a bounded linearoperator (in fact, a contraction) from DT to L(X). For a proof of this, see [5, 9],or Johnson et al., in preparation. (The proof given in the first reference is forthe time independent setting although the proof for the time dependent settingis essentially identical except for the weights used.)

With the disentangling map in hand, we can now address how Theorem 2applies to the improvement of the stability theory of Feynman’s operationalcalculi. We will not do this in an exhaustive way, but will consider a typicalstability theorem for the operational calculus in the time-dependent setting.The theorem we will consider is the following theorem from [18]. (Also statedin the introduction, above.)

Theorem 3 Let Ai : [0, T] → L(X), i = 1, . . . , n, be continuous with respect tothe usual topology on [0, T] and the norm topology on L(X). Associate toeach Ai(·) a continuous Borel probability measure μi on [0, T]. Let {μik}∞k=1,i = 1, . . . , n, be sequences of Borel probability measures on [0, T] such that, foreach i = 1, . . . , n, μik ⇀ μi. Construct the direct sum Banach algebra

UD :=∑

k∈N∪{0}

⊕DT

((A1(·), μ1k)

∼ , . . . , (An(·), μnk)∼)


∼ , . . . , (An(·), μn)∼)

. Then

limk→∞

∣∣�

(T T

μ1k,...,μnk(πk(θ f ))

) − �(T T

μ1,...,μn(π0(θ f ))

)∣∣ = 0

for all � ∈ L(X)∗ and all θ f = ( f, f, f, . . .) ∈ UD.

With Theorem 2 in hand, the statement of the theorem above can bechanged to the following:

Theorem 4 For i = 1, . . . , n, let Ai : [0, T] → L(X), where X is a Banachspace. We assume that each Ai(·) is continuous with respect to the usual topologyon [0, T] and the norm topology on L(X). Associate to each Ai(·) a continuousBorel probability measure μi on [0, T]. For each i = 1, . . . , n, let {μik}∞k=1 be asequence of continuous Borel probability measures on [0, T] such that μik ⇀ μ

as k → ∞. Construct the direct sum Banach algebra

UD :=∑

k∈N∪{0}

⊕DT

((A1(·), μ1k)

∼ , . . . , (An(·), μnk)∼)


∼ , . . . , (An(·), μn)∼)

. Then, forany θ f := ( f, f, f, . . .) ∈ UD and any φ ∈ X, we have

limk→∞

∥∥T T

μ1k,...,μnk(πk(θ f ))φ − T T

μ1,...,μn(π0(θ( f ))φ

∥∥

X= 0 (31)


where πk is the canonical projection of UD onto the disentangling algebraindexed by the measures μ1k, . . . , μnk. Of course, we can think of the conclusionin (31) as

limk→∞

∥∥T T

μ1k,...,μnk( f )φ − T T

μ1,...,μn( f )φ

∥∥

X= 0.

Remark 3 Comparing the statement of Theorem 4 to the statement of Theo-rem 3, we see the difference quite clearly. We obtain strong operator conver-gence in Theorem 4, a much stronger conclusion than in Theorem 3. Indeed, inthe time-dependent setting of the operational calculus that is considered hereand in [16, 17], and [18], only weak convergence results were obtained. In viewof Theorem 2 above, the weak convergence obtained in [16, 17], and [18] canbe changed to strong operator convergence when appropriate assumptions areput on the Banach space in question. It is much more desirable in the setting ofFeynman’s operational calculus to have strong convergence as this is a naturalsetting for the stability questions one considers in relation to the operationalcalculus.

Proof Let m1, . . . , mn ∈ N and let φ ∈ X. We show first that

limk→∞

‖ T Tμ1k,...,μnk

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) φ −

T Tμ1,...,μn

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) φ‖X = 0. (32)

NOTE It is in proving this assertion that the difference in the proof of thistheorem as compared to Theorem 3 arises. We need only choose a vector fromX instead of a linear functional on L(X) as was required in the original proofof Theorem 3.

Using the definition of the disentangling map, remembering that we are inthe continuous measure setting of the operational calculus, we can write thenorm difference above as

∥∥∥∥∥∥

∑

π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ(μ

m11k × · · · × μ

mnnk

)(ds1, . . . , dsm)

−∑

π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ(μ

m11 × · · · × μmn

n

)

× (ds1, . . . , dsm)

∥∥∥∥∥∥

X

. (33)

We now note that, since the operator-valued functions Ai(·) are all continuous,the function fm : [0, T]m → X given by

fm(s1, . . . , sm) = Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ (34)

292 L. Nielsen

is then a continuous function. Moreover, it is a norm-bounded function into theBanach space, since each of the Ai(·) is a continuous function on a compactsubset of R. Also, since [0, T]m is a separable metric space, we have μ

m11k ×

· · · × μmnnk ⇀ μ

m11 × · · · × μmn

n as k → ∞ (see [2], Theorem 3.2). It follows fromTheorem 2 that

limk→∞

∥∥∥∥∥∥∥

∑

π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ(μ

m11k × · · · × μ

mnnk

)(ds1, . . . , dsm)

−∑

π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ(μ

m11 × · · · × μmn

n

)

× (ds1, . . . , dsm)

∥∥∥∥∥∥

X

= 0. (35)

This establishes our assertion.We now sketch the remainder of the proof, reminding the reader that it

follows the proof of Theorem 3.1 of [18] very closely. Let θ f = ( f, f, f, . . .) ∈UD and write f as in (20) above. For φ ∈ X, we can write

‖T Tμ1k,...,μnk

(πk(θ f ))φ − T Tμ1,...,μn

(π0(θ f ))φ‖X

=∥∥∥∥

∞∑

m1,...,mn=0

am1,...,mn

∑

π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ

· (μ

m11k × · · · × μ

mnnk

)(ds1, . . . , dsm)

−∞∑

m1,...,mn=0

am1,...,mn

∑

π∈Sm

∫

�m(π)

Cπ(m)(sπ(m))

· · · Cπ(1)(sπ(1))φ(μ

m11 × · · · × μmn

n

)(ds1, . . . , dsm)

∥∥∥∥

X

� ‖φ‖X

( ∞∑

m1,...,mn=0

|am1,...,mn |∑

π∈Sm

∫

�m(π)

‖Cπ(m)(sπ(m))‖ · · · ‖Cπ(1)(sπ(1))‖

(μ

m11k × · · · × μ

mnnk

)(ds1, . . . , dsm)

+∞∑

m1,...,mn=0

|am1,...,mn |∑

π∈Sm

∫

�m(π)

‖Cπ(m)(sπ(m))‖ · · · ‖Cπ(1)(sπ(1))‖

(μ

m11 × · · · × μmn

n

)(ds1, . . . , dsm)

)

= ‖φ‖X(‖ f‖Dk + ‖ f‖D0

)(36)


where the subscript Dk refers to the kth—disentangling algebra in the directsum algebra UD. (The inequality above is arrived at via the triangle inequalityand the standard Banach algebra inequality ‖xy‖ � ‖x‖‖y‖ which results in aproduct of real-valued and consequently commutative functions. The disen-tangling is then “unraveled” or reversed to obtain the final equality above.)Recall that the norm on UD is

‖ {g�}∞�=1 ‖UD = sup�∈N∪{0}

‖g�‖D�.

Let ε > 0 be given. There is a k0 ∈ N such that ‖θ f ‖UD < ‖ f‖k0 + ε. Using (36)we therefore have

‖T Tμ1k,...,μnk

(π0(θ f ))φ − T Tμ1,...,μn

(π0(θ f ))φ‖X � ‖φ‖X(‖ f‖k0 + ‖ f‖0 + ε

)(37)

We see, then, that a summable scalar-valued dominating function for∞∑

m1,...,mn=0

|am1,...,mn | ∥∥Pm1,...,mnμ1k,...,μnk

(A1(·), . . . , An(·)) φ

−Pm1,...,mnμ1,...,μn

(A1(·), . . . , An(·))φ∥∥

X(38)

is

(m1, . . . , mn) → ‖φ‖X |am1,...,mn |(

rm11,k0

· · · rmnn,k0

+ rm11 · · · rmn

n

)+ ‖φ‖X

ε

2m(39)

where the weights ri,k0 are

ri,k0 =∫

[0,T]‖Ai(s)‖μik0(ds) (40)

and, similarly,

ri =∫

[0,T]‖Ai(s)‖μi(ds). (41)

We can therefore apply the Dominated Convergence Theorem for Bochnerintegrals (see, for example, [3]) and pass the limit on the index k through thesum over m1, . . . , mn. Using (32) we finish the proof. �Acknowledgements The author gratefully thanks the referee for the extremely helpful remarksand suggestions that lead to a significant improvement of this paper.

References

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2. Billingsley, P.: Convergence of probability measures, 2nd edn. Wiley Series in Probability andStatistics, John Wiley and Sons, Inc., New York (1968)

3. Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys, Number 15, American Mathe-matical Society (1977)

4. Feynman, R.P.: An operator calculus having applications in quantum electrodynamics. Phys.Rev. 84, 108–128 (1951)

294 L. Nielsen

5. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators:definitions and elementary properties. Russ. J. Math. Phys. 8, 153–178 (2001)

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8. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators: themonogenic calculus. Adv. Appl. Clifford Algebr. 11, 233–265 (2002)

9. Jefferies, B., Johnson, G.W., Nielsen, L.: Feynman’s operational calculi for time-dependentnoncommuting operators. J. Korean Math. Soc. 38, 193–226 (2001)

10. Johnson, G.W., Lapidus, M.L.: The Feynman Integral and Feynman’s Operational Calculus.Oxford Science Publications, Oxford Mathematical Monographs, Oxford Univ. Press, Oxfordand New York (2000)

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Math Phys Anal Geom (2011) 14:295–320DOI 10.1007/s11040-011-9098-y

On Multifractal Rigidity

Alejandro M. Mesón · Fernando Vericat

Received: 29 December 2006 / Accepted: 28 July 2011 / Published online: 16 August 2011© Springer Science+Business Media B.V. 2011

Abstract We analyze when a multifractal spectrum can be used to recover thepotential. This phenomenon is known as multifractal rigidity. We prove thatfor a certain class of potentials the multifractal spectrum of local entropiesuniquely determines their equilibrium states. This leads to a classificationwhich identifies two systems up to a change of variables.

Keywords Multifractal spectrum · Free energy · Gibbs states

Mathematics Subject Classifications (2010) 37C45 · 37C30

1 Introduction

The multifractal analysis has its genesis in the physical ambient [13, 14]. Inthe study of chaotic behaviors, invariant sets with a complex mathematicalstructure are frequently found. These sets can be decomposed into subsets -with some scaling property. This kind of partition is called a multifractal de-composition. To reveal complete information about these level sets a rigorous

A. M. Mesón · F. Vericat (B)Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB),CONICET–UNLP, La Plata, Argentinae-mail: [email protected]

A. M. Mesóne-mail: [email protected]

A. M. Mesón · F. VericatGrupo de Aplicaciones Matemáticas y Estadísticas de la Facultad de Ingeniería (GAMEFI),UNLP, La Plata, Argentina

296 A.M. Mesón, F. Vericat

mathematical description is needed. A first attempt in this way was to consideran attractor A carrying an invariant measure μ which scales with an exponentα in a scale level r. More specifically in [1, 2] was performed a multifractaldecomposition of the attractor A in sets

Kα = {x : μ (Br (x)) ∼ rα as r → 0} . (1)

where Br (x) denotes the ball of centre r and radius ε.A complete description of the multifractal analysis of invariant measures

was done by Pesin and Weiss in [24]. In that work all the results known untilthat moment about smooth conformal maps were extended. The general ideaof multifractal analysis was introduced in [4] as follows: Given a set X and amap g : X → [−∞, +∞] the level sets

Kα = Kα (g) = {x : g (x) = α} ,

and the decomposition X =(⋃

α

Kα

)∪ Y, where Y is the set in which g is not

defined, are considered. If G is a function defined on sets, and F (α) = G (Kα) ,

then the map F is called the multifractal spectrum specified by the pair (g, G) .

When g (x) is the dimension of the measure Dμ (x) and F (α) the Hausdorffdimension dimH Kα of the set Kα , then this spectrum is called the dimensionmultifractal spectrum. The function F (α) gives a description of the fine-scaleproperty of the part of X where the measure μ is concentrated. The dimensionmultifractal spectrum was previously studied for particular cases in [8, 14] andfurther generalized in the above mentioned articles.

Another interesting example is the local entropies spectrum which is ob-tained with g as the local entropy of a dynamical map f and F (α) as the Bowentopological entropy (for non-compact sets) of the level sets. The Hausdorffdimension and the topological entropy are special cases of “characteristicdimensions” in metric spaces. Thus there is a close relationship between thefields of multifractal analysis and dimension theory of dynamical systems. Theknowledge of adequate dimensions of the multifractal decomposition sets isnecessary to investigate the complexity of them.

The classification of multifractal spectra is done by using families of mea-sures {μα}α∈R such that μα (Kα) = 1. Two multifractal spectra (g1, G1) and(g2, G2) are said to be equivalent with respect to the families of full measures{μ1

α

}α∈R and

{μ2

α

}α∈R if there is a bijection σ : [−∞, +∞] → [−∞, +∞] such

that μ1α = μ2

σ(α) for every real α. When the spectrum is defined from a potentialϕ : X → R and dynamics f : X → X, like the entropies spectrum, a one-parameter family of measures

{μq

}q∈R is introduced as the Gibbs state for each

member of a certain family of potentials{ϕq

}. Then a parametrization α (q)

with μq(Kα(q)

) = 1 and μq (Kα) = 0 if α �= α (q) is defined. Therefore, there isa correspondence between the level sets of the decomposition and the family

On Multifractal Rigidity 297

of full measures{μq

}. The parametrization is obtained by setting α (q) :=

−T′(q), where T is the “free energy” in Ruelle’s thermodynamic formalism

terminology, whereas q is interpreted as the inverse of the temperature, so α

may be the internal energy per volume. In the most known and used spectra(for instance the dimension, entropy or the Lyapunov spectra), the free energymap is, under certain conditions, a convex differentiable map whose Legendretransform is F (α) , thus multifractal spectra can be classified by the dynamicsand equilibrium states.

One interesting problem is to study when the spectrum determines thepotential, a phenomenon called multifractal rigidity. In other words the issueis to analyze when the multifractal classification works as a complete invariantof dynamical systems as well as of equilibrium states. This classification fitsbetter to a physical interpretation than the topological and measure-theoreticones, because multifractal classification identifies two systems up to a bijectionbetween variables.

A remarkable result in this direction was obtained in [4], where the authorsestablished multifractal rigidity for the full shift in two symbols and forspecial potentials. Specifically they proved that if two Bernoulli schemes, withprobabilities pi, pi, i = 1, 2, have the same dimension spectrum, then there isa homeomorphism between the respective phase spaces and the probabilitiesare uniquely determined by each multifractal spectrum

A meaningful step was then done by Pollicott and Weiss [25] who demon-strated that for the special class of generic locally constant potentials the freeenergy determines the potential. By locally constant potentials it must beunderstood those that depend on a finite number of coordinates, or finite rangeobservables in the physical language. The genericity is a matrix property, whichmust be verified by the matrices associated to the potentials. The matrices withthis property are in the complement of an algebraic variety of dimension one.In the above mentioned article examples of systems with locally potentialswhich have the same free energy but non-equivalent were presented. Alsothey established a local multifractal rigidity for symbolic dynamical systemsand H

..older continuous potentials.

In this article we establish the existence of multifractal rigidity forlarger classes of potentials than in the mentioned articles. If (X, d) is acompact metric space and f : X → X an homeomorphism the local en-tropies spectrum is given from the decomposition Kα = {

x : hμ (x, f ) = α}

where hμ (x, f ) is the local entropy: hμ (x, f ) = limε→0

lim supn→∞

− 1n log μ

(Bn,ε (x)

),

with Bn,ε (x) the ball of centre x and radius ε in the metric dn (x, y) =max

{d(

f i (x) , f i (y)) : i = 0, 1, ..., n − 1

}. The map E (α) defined on level sets

is E (α) = htop ( f, Kα), with htop ( f, Z ) the Bowen topological entropy for non-compact nor invariant sets [6], and the free energy for this spectrum withpotential ϕ is the map T(q) = P(qϕ) − qP(ϕ) (P the topological pressure)whose Legendre transform is precisely E (α). The description of this mul-tifractal spectrum for a class of potentials broader than Hölder continuous


maps and for expansive homeomorphisms with specification was performedby Takens and Verbitski [30]. The lattice spin systems used in classicalStatistical Mechanics are mathematically modelled by the Markov systems�A = {

x = (xi)i∈Z : xi ∈ �, ∀i ∈ Z, Axi,xi+1 = 1}, where A is a k × k matrix

with 0, 1 entries and � = {0, 1, 2, ..., k − 1} . The integers i are called the sitesand the corresponding coordinate xi the spin at the site i. The matrix Aindicates which configurations, i.e. which sequences x = (xi)i∈Z , are allowed.

We prove, for Markov systems and an adequate class of potentials depend-ing on infinite coordinates, the following result: Eϕ1

= Eϕ2implies μϕ1

= μϕ2,

where μϕ is the Gibbs state associated to the potential ϕ. We use an approachbased on transfer operators which also works for spin lattice models withinfinite range potentials, i.e in which the potential depends on all the coordi-nates, The scheme followed is similar than [25], where stochastic matrices areused. We firstly prove that the multifractal spectrum determines the Fredholmdeterminant of the corresponding transfer operator (it plays the role of the ma-trix in the finite range case), then for the special class of potentials consideredthe determinant is related with the zeta function associated to the potentialand finally since the zeta function uniquely determines the equilibrium stateswe are done. This last result will be actually proved in a much general contextthan symbolic systems.

We also study the variational properties of perturbations on the localentropies spectrum in order to get a local rigidity result. For this we shallconsider for a fixed dynamical map f : X → X a family of potentials � ={ϕλ}λ∈(−δ,δ) and study the variation of the entropies spectrum, by computingthe first and second derivatives, with respect to the perturbative parameter λ,of the function τ (λ, q) := P (qϕλ) − qP (ϕλ) which is in turn a perturbation ofthe function T (q) = P (qϕ0) − qP (ϕ0) . The estimate of the influence of theperturbations and how numerical results could be affected by small perturba-tions is very useful for numerical computations. Results about first variationalformulae for dimension spectra were obtained in [3] and [33] and for thesecond variation, also for dimension spectrum, in [15]. In all these cases theresults are valid for hyperbolic diffeomorphisms. In [20] were calculated thefirst derivative of τ (λ, q) but under much weaker hypothesis than hyperbolic-ity and Hölder continuous potentials, we compute here the second derivativesof τ (λ, q) under these same hypothesis. The local rigidity result that wepresent herein is: If λ → Eϕ

λis constant for λ ∈ (−δ, δ) then μϕλ

is constant forλ ∈ (−δ, δ) , whose validity is established for expansive homeomorphism withspecification, conditions much weaker than the existence of Markov partitions,and for a class which includes on hyperbolic sets.

2 Basic Definitions and Previous Results

We begin by recalling the description of the local entropies multifractalspectrum, some of whose main aspects were sketched in the introduction: Let(X, d) be a compact metric space, and f : X → X a continuous map. Let


dn (x, y) = max{d(

f i (x) , f i (y)) : i = 0, 1, ..., n − 1

}. We denote by Bn,ε (x)

the ball of centre x and radius ε in the metric dn. If μ a f−invariant measure,the upper and lower local entropies are

hμ (x, f ) = limε→0

lim supn→∞

− 1n

log μ(Bn,ε (x)

)

hμ (x, f ) = limε→0

lim infn→∞ − 1

nlog μ

(Bn,ε (x)

).

Then (Brin–Katok theorem [7]), the local entropy does exist, i.e. hμ (x, f ) =hμ (x, f ) := hμ (x, f ), for μ − a.e. x ∈ X. Now the local entropies spectrum of

f is that specified by the pair(hμ ( f, x) , E (α)

)with E (α) := htop ( f, Kα) . The

set X is naturally decomposed as

X =∞⋃

α=−∞Kα ∪ (X − Y) ,

where Y is the set in which the local entropy map does not exist and isusually called the irregular part of the spectrum. By the Brin–Katok theoremμ (X − Y) = 0, for any f−invariant measure μ.

Next we collect a few definitions from the Ruelle thermodynamic formalism[26]. The topological pressure associated to f and to a potential ϕ : X → R, isthe number

P (ϕ) = supμ

{hμ ( f ) +

∫ϕdμ

},

where the supremum is taken over all the f−invariant Borel measures μ onX, and hμ ( f ) is the usual Kolmogorov measure-theoretic entropy of f.

An equilibrium state for the potential ϕ is a measure μϕ for which:

P (ϕ) = hμϕ( f ) +

∫ϕdμϕ. (2)

The set of equilibrium states for the potential ϕ will be denoted by Mϕ (X) .

Under certain conditions imposed on the map f and the potential ϕ an equi-librium state can be constructed [16–30]. The specif ication property for a mapf : X → X intuitively says that for specified orbit segments a periodic orbitapproximating the trajectory can be found. This condition ensures abundanceof periodic points. It is a concept introduced by R. Bowen [5]. Formally, ahomeomorphism f : X → X has the specif ication property if given a finitedisjoint collection of integer intervals I1, I2..., Ik and ε > 0, there is an integerM (ε) and a function � : I = ∪Ii → X, such that the following conditions aresatisfied:

(i) dist(Ii, I j

)> M (ε) (Euclidean distance)

(ii) f n1−n2 (� (n1)) = �(n2)

(iii) d ( f n (x) , � (n)) < ε, for some x : f m (x) = x, with m ≥ M (ε) +length (I) and for every n ∈ I.


A homeomorphism f : X → X is called expansive if there is a constant δ >

0, such that d ( f n (x) , f n (y)) < δ, for any integer n implies x = y.

For a potential ϕ we put

Sn (ϕ) (x) =n−1∑i=0

ϕ(

f i (x))

(3)

which is called the statistical sum.Following [16] or [30], we say that a potential ϕ belongs to the class ν f (X)

if it satisfies the following condition:There are constants ε, K > 0 such that

dn (x, y) < ε =⇒ |Sn (ϕ) (x) − Sn (ϕ) (y)| < K. (4)

We also recall how an equilibrium state associated to a potential ϕ ∈ ν f (X)

can be defined. Let Pn ( f ) = {x : f n (x) = x}, then we set

μϕ,n(A) = 1Z ( f, ϕ, n)

∑x∈Pn( f )

exp (Sn (ϕ) (x)) δx(A), (5)

where Z ( f, ϕ, n) = ∑x∈Pn( f )

exp (Sn (ϕ) (x)) and δx is the Dirac measure at x:

δx(A) ={

1 if x ∈ A0 if x /∈ A

.

If X is compact the sequence{μϕ,n

}has an accumulation point and under

the above conditions it has a weak limit μϕ, i.e.:

limn→∞

∫ψ (x) dμϕ,n =

∫ψ (x) dμϕ, (6)

for every continuous ψ [16, 26].

Theorem [16, 26] Let f be an expansive homeomorphism with thespecif ication property and ϕ a potential belonging to the class ν f (X) , then μϕ

is the unique equilibrium state associated to ϕ. Besides μϕ is ergodic.

The conditions of expansiveness and specification are much weaker than theexistence of Markov partitions. Under these hypothesis Takens and Verbitski[30] developed a multifractal formalism for local entropies spectrum, wereview here the main results: let T(q) = P(qϕ) − qP(ϕ), q ∈ R, called the freeenergy of ϕ,

(i) the function T(q) is convex and continuously differentiable. This maphas a Legendre transform E (α) = inf

q∈R{qα − T (q)}. E (α) describes local

entropies spectrum f .(ii) If Kα = {

x : hμϕ(x, f ) = α

}, (μϕ �= μmax, the measure maximal entropy),

then E (α) = htop ( f, Kα). Besides

E (α (q)) = qα (q) + T (q) ; α (q) := −T′(q) , q = E ′ (α) . (7)


Let αi = limq→∞ α (q) = inf

q∈R{α (q)} , αs = lim

q→−∞ α (q) = supq∈R

{α (q)} , then Kα =∅, if α /∈ (αi, αs) , so that the domain of definition of E (α) is the range of T

′(q) .

Definition A f -invariant measure μ is a Gibbs state if for sufficiently smallε > 0, there are constants Aε, Bε > 0, such that for any x ∈ X and for anypositive integer n:

Aε (exp (Sn (ϕ)(x))−nP (ϕ)) ≤ μ(Bn,ε (x)

)≤ Bε (exp (Sn (ϕ)(x))−nP (ϕ)) , (8)

where Sn (ϕ) (x) =n−1∑i=0

ϕ(

f i (x)).

Theorem [16–26] Let f : X → X be an expansive homeomorphism which havethe specif ication property and ϕ a potential belonging to the class νf (X), then μϕ

is an equilibrium state associated to ϕ, which is a Gibbs state. Besides it is ergodic.

The multifractal spectrum of local entropies is thus described by the familyof measures

{μq

}whose members are the Gibbs states associated to the

potentials qϕ − qP(ϕ). One has μq(Kα(q)

) = 1, with α (q) = −T′(q) .

One important general result about classification of equilibrium states is thefollowing:

Theorem [30] Let ϕ, ψ ∈ ν f (X) where X is a compact metric space and f anexpansive homeomorphism with specif ication, then μϕ = μψ if and only if thereis a constant C such that Sn (ϕ) (x) = Sn (ψ + C) (x) , for any n and for everyx ∈ Pn ( f ) = {x : f n (x) = x} .

A proof of the above claim for the particular case of hyperbolic systems inRiemannian manifolds and Hölder continuous potentials. appears in [16, pp.636–637].

According to the nomenclature of [25] the unmarked orbit spectrum, theweak orbit spectrum and the marked periodic spectrum of the potential ϕ arerespectively

Sϕ = {(Sn (ϕ) (x) , n) : x ∈ Pn ( f )} ,

Wϕ = {Sn (ϕ) (x) : x ∈ Pn ( f )} ,

Hϕ = {(Sn (ϕ) (x) , x) : x ∈ Pn ( f )} .

In [25] is made an interesting parallelism between these spectra and lengthspectra of geodesics in compact hyperbolic surfaces. For instance Sϕ is theanalogue of the unmarked length spectrum which consists of the length of allclosed geodesics and Wϕ is the analogue of the set of the lengths of all closedgeodesics marked with the free homotopy class of the geodesic. In this way isestablished a comparison between multifractal rigidity and the Kac problemcan you hear the shape of a drum?, a question which summarizes the problemabout when the geodesic spectrum determines the manifold.


A “Hamiltonian” approach to the presented multifractal rigidity can beformulated as follows: let f : X → X be an expansive homeomorphism withthe property of specification and a potential ϕ in the class ν f (X), so that it hasa Gibbs state μϕ. In [19] we have introduced a Hamiltonian of the form

Hn,ε(x) = − log μϕ

[Bn,ε (x)

]This Hamiltonian may be considered as a generalization to the Sinai’s one [29].In that case the measure is the probability associated with cylinders. It shouldbe noticed that balls like Bn,ε (x), in the particular case of symbolic spaces (witha certain metric), correspond to cylinders.

Physically the point x in the Hamiltonian can be thought as a microstatewhose energy is given by the interaction of the point x with all the points ofthe ball Bn,ε (x), i.e. with all the points that follows the trajectory of x withinε-distance up to time n. The total interaction being given by the measure ofthe ball. The microstates we are interested in are the whole set of periodicpoints Pn ( f ). In analogy with statistical mechanics, we introduce the canonicalpartition function (q interpreted as the inverse of the temperature):

Z (q; n, ε) =∑

x∈Pm( f )

exp[−qHn,ε(x)

] =∑

x∈Pn( f )

(μϕ

[Bn,ε (x)

])q (9)

and a “free energy”

F(q) = limε→0

limn→∞

1n

log Z (q; n, ε). (10)

We have proved [19] that F(q) = P(qϕ) − qP(ϕ), for every real q. So thatthis energy function agrees with that introduced by Takens and Verbitski fortheir multifractal formalism which will be used for.

3 Local Multifractal Rigidity

Let us begin considering a homeomorphism f : X → X, with X a compactmetric space, and a Ck− family of potentials � = {ϕλ}λ∈(−δ,δ) ⊂ ν f (X) seen asa perturbation of a fixed potential ϕ0. The requirement for the value of k willdepend of the order of derivative that we wish to compute. Next we introducethe map τ (λ, q) = P(qϕλ) − qP(ϕλ), where P = P (ϕλ, f ) , λ ∈ (−δ, δ) . Forthe non-perturbed case the map τ (0, q) = P(qϕ0) − qP(ϕ0) will be denoteddirectly by T(q), the free energy of ϕ0.


The following results were set in [20]:

(1) ∂τ(λ,q)

∂λ|λ=0= q

∫�dμq, where � := ∂ϕλ

∂λ|λ=0 − ∂ P(ϕλ)

∂λ|λ=0= ∂ϕλ

∂λ|λ=0 −∫

∂ϕλ

∂λ|λ=0 dμq and μq is an equilibrium state for qϕ0.

(2) If μ0 := μϕ0�= μmax, then μϕλ

�= μmax for sufficiently small |λ| .

The necessity of establishing a result of this nature is to ensure thedifferentiability of the map λ → Eλ (α). Indeed if μ = μmax, then [30] E (α) =htop ( f, Kα) =

{htop if α = htop

0 if α �= htop. Now it must be checked that under small

perturbations one cannot have this degenerate behavior if it does not occur inthe non-perturbed case.

Lemma 1 Let ϕ, ψ ∈ ν f (X), with f a homeomorphism with specif ication, thend2 P(ϕ+λψ)

dλ2 |λ=0= Cϕ(ψ) := μϕ(ψ2) − (μϕ(ψ))2, seeing the measure as a func-tional by μϕ(ψ) = ∫

ψdμϕ , and μϕ the Gibbs state associated to ϕ.

Proof By the multifractal formalism described in the earlier section: dP(ϕ+λψ)

dλ

|λ=0= μϕ (ψ) = ∫ψdμϕ. Let us denote μλ = μϕ+λψ and so we have dP(ϕ+λψ)

dλ=

μλ (ψ) . Let us recall (c.f. (5) and (6)) that the Gibbs state for a potential ϕ inthe class ν f (X) is defined as the weak limit μϕ of the “Gibbs ensembles”

μϕ,n ({y}) = exp (Sn (ϕ) (y))∑x∈Pn( f )

exp (Sn (ϕ) (x)). (11)

By the compactness of X this sequence has an accumulation point whichcan be interpreted as its “thermodynamic limit”. Thus for obtaining thesecond derivative we must differentiate μn,λ (ψ). Doing this we have dμn,λ(ψ)

dλ

=∑

x∈Pn( f )ψ2 exp(Sn(ϕ+λψ)(x))∑

x∈Pn( f )exp(Sn(ϕ+λψ)(x))

−[ ∑

x∈Pn( f )ψ(x) exp(Sn(ϕ+λψ)(x))∑

x∈Pn( f )exp(Sn(ϕ+λψ)(x))

]2

and then d2 P(ϕ+λψ)

dλ2 |λ=0=μϕ(ψ2) − (μϕ(ψ))2. ��

Theorem 1 Let � = {ϕλ}λ∈(−δ,δ) ⊂ ν f (X) be a C2-family, with f a homeomor-phism with the specif ication property, then

∂2τ (λ, q)

∂λ2 |λ=0= q[μqϕ0

(∂2ϕλ

∂λ2 |λ=0

)− μϕ0

(∂2ϕλ

∂λ2 |λ=0

)]

+ q2[

Cqϕ0

(∂ϕλ

∂λ|λ=0

)− Cϕ0

(∂ϕλ

∂λ|λ=0

)].


Proof We start by calculating d2 P(ϕ+λψ)

dλ2 |λ=0, where {ψλ}λ∈(−δ,δ) is a C2−family.For this we must differentiate μϕ+λψλ,n (ψλ) with respect to λ. Thus

μϕ+λψλ,n (ψλ) =∑

x∈Pn( f )ψλ exp(Sn(ϕ+λψλ)(x))∑

x∈Pn( f )exp(Sn(ϕ+λψλ)(x))

, and so

dμϕ+λψλ,n(ψλ

)dλ

=

∑x∈Pn( f )

[∂ψλ

∂λexp (Sn (ϕ + λψλ) (x)) + ψλ exp (Sn (ϕ + λψλ) (x))

(λ

∂ψλ

∂λ+ ψλ

)]∑

x∈Pn( f )exp (Sn (ϕ + λψλ) (x))

−

( ∑x∈Pn( f )

ψλ exp (Sn (ϕ + λψλ) (x))

) ∑x∈Pn( f )

exp (Sn (ϕ + λψλ) (x))

(∂ψλ

∂λ+ λψλ

)[ ∑

x∈Pn( f )exp (Sn (ϕ + λψλ) (x))

]2

evaluating in λ = 0

∑x∈Pn( f )

[∂ψλ

∂λ|λ=0 exp (Sn (ϕ) (x))

]∑

x∈Pn( f )exp (Sn (ϕ) (x))

+

∑x∈Pn( f )

[ψ2

0 exp (Sn (ϕ) (x))]

∑x∈Pn( f )

exp (Sn (ϕ) (x))

−

∑x∈Pn( f )

[ψ0 exp (Sn (ϕ) (x))

]∑


×

∑x∈Pn( f )

[ψ0 exp (Sn (ϕ) (x))

]∑


= Cϕ (ψ0) + μϕ

(∂ψλ

∂λ|λ=0

).

Now

∂2τ (λ, q)

∂λ2 |λ=0= ∂2 P (qϕλ)

∂λ2 |λ=0 −q∂2 P (ϕλ)

∂λ2 |λ=0

=∂2 P

(qϕ0 + q

∂ϕλ

∂λ|λ=0 λ + q

∂2ϕλ

∂λ2 |λ=0 λ2 + o(λ2

))∂λ2 |λ=0

− q∂2 P

(ϕ0 + q

∂ϕλ

∂λ|λ=0 λ + q

∂2ϕλ

∂λ2 |λ=0 λ2 + o(λ2

))∂λ2 |λ=0

= q[μqϕ0

(∂2ϕλ

∂λ2 |λ=0

)− μϕ0

(∂2ϕλ

∂λ2 |λ=0

)]

+ q2[

Cqϕ0

(∂ϕλ

∂λ|λ=0

)− Cϕ0

(∂ϕλ

∂λ|λ=0

)].

��


If we define a map D (λ) by P (D (λ)ψλ) = 0, then we can calculate from the

above theorem∂2 D (λ)

∂λ2 |λ=0 . The interest of such a computation resides in the

fact that, for the particular case of hyperbolic systems with basic set � we have,by the Bowen equation, dimH � = D(0) (dimH means Hausdorff dimension).Thus we can find a first and a second variational formula for a “like perturbeddimension” under the general hypothesis of Theorem 1. A formula of this stylewas supplied in [15], but under stronger conditions.

Proposition 1 Under the same conditions for the dynamics and the potential as

in Theorem 1 and for D(λ) def ined as above, it holds: ∂ D(λ)

∂λ|λ=0= −D(0)μ(

∂ψλ∂λ

|λ=0)

μ(ψ0)

and ∂2 D(λ)

∂λ2 |λ=0= {−CD(0)ψ0(ψ0

∂ D(λ)

∂λ|λ=0 +D(0)

∂ψλ

∂λ|λ=0) × −2 ∂ D(λ)

∂λ|λ=0 ×

μ(∂ψλ

∂λ|λ=0) −D(0)μ(

∂2ψλ

∂λ2 |λ=0)} × 1μ(ψ0)

, where μ is the Gibbs state associated toD(0)ψ0.

Proof We have 0 = ∂ P(D(λ)ψλ)

∂λ|λ=0= μD(0)ψ0(

∂(D(λ)ψλ)

∂λ|λ=0) = μD(0)ψ0×

(ψ0∂ D(λ)

∂λ|λ=0 +D(0)

∂ψλ

∂λ|λ=0), and so ∂ D(λ)

∂λ|λ=0= −D(0)μD(0)ψ0 (

∂ψλ∂λ

|λ=0)

μD(0)ψ0 (ψ0).

For the second derivative formula 0 = ∂2 P(D(λ)ψλ)

∂λ2 |λ=0=μD(0)ψ0(

∂2(D(λ)ψλ)=∂λ2 |λ=0) + CD(0)ψ0

(∂2(D(λ)ψλ)=

∂λ2 |λ=0) = μD(0)ψ0(ψ0∂2(D(λ)ψλ)

∂λ2 |λ=0 +2 ∂ D(λ)

∂λ|λ=0

∂ψλ

∂λ|λ=0 +D(0)

∂2ψλ

∂λ2 |λ=0) + CD(0)ψ0ψ0

∂ψλ

∂λ|λ=0 D(0)

∂ψλ

∂λ|λ=0. So that

∂2 D(λ)

∂λ2 |λ=0= −CD(0)ψ0(ψ0

∂ D(λ)

∂λ|λ=0 +D(0)

∂ψλ

∂λ|λ=0)−2 ∂ D(λ)

∂λ|λ=0 ×μ(

∂ψλ

∂λ|λ=0)−

D(0)μD(0)ψ0(

∂2ψλ

∂λ2 |λ=0). ��

Finally we state our result of local multifractal rigidity

Theorem 2 Let f : X → X be an expansive homeomorphism in a compactmetric space with the specif ication property. Let � = {ϕλ}λ∈(−δ,δ) ⊂ ν f (X) bea C2-family such that Eϕλ

(α) is constant, then μϕλis also constant.

Proof From the equality of the multifractal spectra we deduce that the mapλ → τ (λ, q) is constant, for each q and for |λ| < δ. Therefore

∫�λdmλ,q = 0,

where �λ := ∂ϕλ

∂λ− ∂ P(ϕλ)

∂λand with mλ,q the Gibbs state associated to ϕ

λ,q =qϕλ − P(ϕλ), λ ∈ (−δ, δ) .

Let us recall the classical Mazur theorem about existence of tangent func-tionals in Banach spaces [9, p. 450]: if V is a separable Banach space andP : V → R is convex continuous then the set at which there is a unique tangentfunctional to P contains a countable intersection of dense open sets, andso, because V is a Banach space, by the Baire category theorem it is dense.This theorem can be applied with P : C (X) → R the topological pressureand the tangent functionals at ϕ defined as the set of the signed measuresμ such that P (ϕ + ψ) − P (ϕ) ≥ ∫

ψdμ, for any ψ ∈ C (X) . If the entropymap μ → hμ ( f ) is upper semi-continuous then the set of tangent functionals


at ϕ agrees with the set of equilibrium states of ϕ and if f is an expansivecontinuous map in a compact metric space then the entropy map is uppersemi-continuous [32]. Now under the hypothesis considered in this work itholds that there is a dense subset A of C (X) such that for any ϕ ∈ A the setMϕ (X) has just one element. Based upon the above results we have that d

dλ

∫(ϕλ − P(ϕλ))dμ = 0 for any equilibrium measure μ associated to potentials inan open dense subset of ν f (X) . Thus

∫ϕλ − P(ϕλ)dμ = C, for some constant

C and so Sn (ϕλ − P(ϕλ)) (x) = Sn (Cλ) (x) is a small neighborhood of λ = 0.

Therefore there is a small interval (−δ, δ) such that μϕλ−P(ϕλ) = μϕλis constant

for λ ∈ (−δ, δ) . ��

The above proposition generalizes a similar result of [4]. There was proveda local multifractal rigidity theorem, but for hyperbolic systems and for thedimension spectrum instead.

4 Multifractal Rigidity for Spin Lattice Systems

The next step is to address to the following multifractal rigidity problem: letEϕi

(α) , i = 1, 2 be two multifractal spectra of local entropies defined frompotentials ϕi which an unique associated Gibbs state and dynamics f : X → X.

Under adequate conditions these spectra are determined by the respective freeenergies Tϕi

(q) since they are the Legendre transforms of Eϕi(α). Now the

problem will be to find classes of potentials and dynamics for which the freeenergy determines the equilibrium states. In short the issue is to establish whenthe following implication is valid

Eϕ1 (α) = Eϕ2 (α) =⇒ μϕ1 = μϕ2 . (12)

We briefly describe the special case treated by Barreira et al. in [4]: theyhave considered a one-dimensional map f : I → I (I = [0, 1]) which can be“partitioned” in two maps fi : Ii → Ii, i = 1, 2, with Ii ⊂ [0, 1] and fi (Ii) =[0, 1] . If J =

∞⋂k=1

f −k (I1 ∪ I2) then {J ∩ I1, J ∩ I2} is a Markov partition for

(J, f ) and this dynamical system is topologically conjugated to the full shiftof two symbols �2 = {x = (xi)i∈N : xi ∈ {0, 1}}, which is a Bernoulli schemewith probabilities pi, i = 0, 1, assigned to each xi. The potential is ϕ : �2 → Rdefined by ϕ (x) = log pi if x ∈ Ii, this map is in fact of the form ϕ (x) = ψ (x0)

with the probabilities pi = exp(ψ(i))exp(ψ(0))+exp(ψ(1))

, i = 0, 1, while the topological pres-sure is P (ϕ) = log (exp (ψ (0)) + exp (ψ (1))) . Thus a direct calculation leads toTϕ (q) = log

(∑pq

i

). The Gibbs state associated to ϕ is the product measure in

�2 of the measures pi.

Let f, f be one-dimensional Markov maps with invariant sets J, J as aboveand let χ, χ : �2 → R be the coding maps giving the conjugations betweeneach J, J and �2. In [4] it is then proved that there is a homeomorphism φ :J → J such that φ ◦ f = f ◦ φ, so that the dynamical systems (J, f ) and

(J, f

)are topologically conjugated, and there is an automorphism ρ of �2 such that


κ ◦ φ = ρ ◦ χ . This was established by showing that the free energy uniquelydetermines the probabilities. Now in this special situation (12) holds.

As we mentioned in the introduction the problem on whether the free en-ergy determines the potential was solved by Pollicott and Weiss for potentialsdepending on a finite number of coordinates (finite range potentials). Our aimherein is to establish the validity of (12) for a class which include infinite rangepotentials, i.e. depending on the entire configuration. One interesting examplein this situation is the Kac model: let � = {±1} with the transition matrix with

all entries equal to 1 and the potential ϕ (x) = Jx0

∞∑n=1

xnλn, with λ ∈ (0, 1) ,

J ∈ R is a coupling parameter.In the case of finite range potentials can be defined a primitive matrix

(H is primitive if exists a positive integer p such that H p has all its entriespositive). Indeed if ϕ : �A → R depends on two coordinates let Lϕ = Li, j ={

0 if Ai, j = 0exp ϕ (x) if Ai, j = 1 , with x0 = i, x1 = j, for instance in the Ising model

ϕ (x) = Jx0x1 and Li, j = exp(Jxix j

). If we consider a “partition function”

Zn (ϕ) = ∑x∈Pn(σ )

exp (Sn (ϕ) (x)) then

Zn (ϕ) = Tr(Ln) . (13)

On the other hand the “thermodynamic limit” limn→∞

1n log Zn (ϕ) does exist and

equals logE1 (L) , where E1 is the leading positive eigenvalue of L [26]. Theexistence of such a leading eigenvalue is ensured by the Perron–Frobeniustheorem, since the matrix is primitive. For Hölder continuous potentials isvalid P (ϕ) = lim

n→∞1n log Zn (ϕ) [16].

If we are in the more general situation of not having always potentialsdepending on a finite number of coordinates we must work with other classof objects than matrices. They will be transfer operators, in the style of thoseintroduced by Ruelle in his thermodynamic formalism, and the aim will be toobtain an analogous relationship to (13) with the trace of the operator insteadof the matrix.

Next we shall write down such an operator: for

�+A = {

x = (xi)i∈N : xi ∈ �, ∀i ∈ N, Axi,xi+1 = 1}

and ϕ ∈ C(�+

A

), let

Lϕ (κ) (x) =∑i∈�

Ai,κ0 exp (ϕ (i, x)) χ ((i, x)) , (14)

where (i, x) is the configuration(i, x0, x1,...

). The space of finite range poten-

tials, i.e. depending on a finite number of coordinates, is left invariant by L andso the operator can be reduced in this subspace to a matrix like L for which therelationship (13) is satisfied.


Let us return to the Kac model, in this case the transfer operator reads:

Lϕ (κ) (x) =∑i=±1

exp

(Jx0

∞∑n=1

xnλn

)χ ((i, x)) . (15)

Next we consider the space of functions A∞(�+A) := {ϕ ∈ C(�+

A) : there existsa χ ∈ A∞(DR) with ϕ(x) = χ(π(x))}, where DR = {z : |z| = R} and π is a

projection π : �+A → DR defined by the assignation x −→

∞∑n=1

xn−1λn. The

space A∞ (U) is the space of complex functions holomorphic in U and contin-uous in U (the closure of U), endowed with the norm ‖χ‖ = sup

z∈DR

|χ (z)|. On

A∞(�+

A

)the operator Lϕ induces another one acting on A∞ (DR), also denoted

by Lϕ, in the following way:Let ψ j : DR → DR, ψ j (z) = λ ( j + z) , j = ±1, and thus

Lϕ (κ) (z) =∑j=±1

exp (Jxz) χ(ψ j (z)

), (16)

for χ ∈ A∞ (DR) .

By using the trace formula deduced from [17] we have

Zn (ϕ) = (1 − λn) Tr

(Ln

ϕ

) = Tr(Ln

ϕ

) − Tr(Ln

ϕ

), with L = λL, (17)

what we were looking for, i.e. a relationship in the style of (13) with theoperator playing the role of the matrix.

Now the task will be to develop a more general approach to obtain a similarresult. For this we shall work in spin lattice systems modeled by finite subshifttype

(�+

A, σ)

with potentials ϕ : �+A → R for which the following conditions be

satisfied:

(C1) There is a projection π : �+A → Rd, for some d ≥ 1, and open sets

{Wi} ⊂ Rd such that π(�+

A

) ⊂ ⋃i

Wi and maps ψi : ⋃j∈�i

W j → Wi (� j :={i ∈ � : Ai, j = 1

}. Besides π (i, x) = ψi (π (x)) ∈ �+

A, recall that (i, x) isthe configuration

(i, x0, x1,...

).

(C2) There are neighborhoods Ui ⊂ Cd of Wi such that each ψi extendsholomorphically to

⋃j∈�i

U j and applies⋃j∈�i

U j strictly inside itself. By

“strictly inside itself” we understand: let D be a bounded connectedsubspace of a Banach space B and ψ a holomorphic map defined on D.

We say that ψ applies D strictly inside itself if infz∈D, z′ ∈B− D

∥∥ψ (z) − z′∥∥ ≥

δ > 0.

(C3) There exists holomorphic functions ϕi defined on Ui such that ϕ (i, x) =ϕi (ψi (π (x))) , for any x ∈ �+

A.


These conditions allow to define a transfer operator by:

Lϕ :⊕i∈�

A∞ (Ui) →⊕i∈�

A∞ (Ui) (18)

(Lϕ (χ)

)i (z) =

∑j∈�

Ai, j exp(ϕ j

(ψ j (z)

))χ(ψ j (z)

)

A trace formula for such an operator, in the style of the Atiyah–Bottformula on Lefschetz fixed point, is displayed in [17] as:

Tr(Lϕ

) =∑i∈�

Ai,i exp (ϕi (zi))1

det (1 − Dψi (zκ)), (19)

where zi is the fixed point of ψi and Dψ is the differential map of ψ, seen asa linear operator. It must be pointed out that, by the Earle–Hamilton theorem[10] a map ψ applying strictly a domain D inside itself has exactly a fixed pointz ∈ D with ‖Dψ (z)‖ < 1.

A relevant fact about these transfer operators is that they are nuclear. Letus recall that an operator L acting on a Banach space B is nuclear if there existsequences (xn) ⊂ B, ( fn) ⊂ B∗ (the dual space of B) with ‖xn‖ = 1, ‖ fn‖ =1 and numbers (ρn) with

∞∑n=0

|ρn| < ∞ such that L (x) =∞∑

n=0ρn fn (x) xn for

every x ∈ B. The nuclearity of operators similar to (18) and also for thosecorresponding to a continuous case was established in [21, 22]. These proofscan be easily adapted to operators (18) and so we will omit it.

Let us consider now the family of operators Lq, which are the transferoperators associated to the family of potentials {qϕ}. In this case the condition(C3) is formulated as follows: there exist holomorphic functions ϕi,q defined onUi such that qϕ (i, x) = ϕi,q (ψi (π (x))) , for any x ∈ �+

A. These operators willbe denoted by Lq.

By the Grothendieck theory for nuclear operators [11, 12] the Fredholmdeterminant det(1 − zLq) is an entire map in both variables z, q and it has

the expansion det(1 − zLq) = exp(−∞∑

n=1

zn

n Tr(Lnq)). If the charts ψi, defined in

(C1)–(C3) are constant then by the Mayer trace formula it holds Zn(q) :=Zn(qϕ) = Tr(Ln

q), this is the case, for instance, for the Ising model and manyother statistical systems. If the ψi are linear, like in the Kac-model, there is alsoa relationship between the partition function Zn(q) and the trace of Ln

q in thestyle of (17). The general relationship between partition function and trace is

Zn (q) =d∑

p=0

Tr[(

L(p)q

)n], (20)


where L(p)q are operators defined on

⊕κ∈�

∧pB (Uκ) , where

∧pB (Ui) is the space

of the differential p-forms holomorphic on Ui, as

L(p)q :

⊕i∈�

∧p

B (Ui) →⊕i∈�

∧p

B (Ui) , Ui ⊂ Cd

(L(p)

q

(wp

))i(z) =

∑j∈�

Ai, j exp(ϕ j,q (z)

)∧p

Dψ j (z)(wp

) (ψ j (z)

),

here wp ∈ ∧pB (Ui) and

∧p

Dψ is the p-fold exterior product of the differential

map Dψ (considered as a linear operator). We have L(0)q = Lq and any L(p)

q isnuclear, as a natural extension of the fact that L(0)

q does. Thus the Fredholmdeterminant Dp(z, q) := det(1 − zL(p)

q ) is entire in z and q, for any p.

Now for p = 0, d = 1 and constant charts there is an obvious and directrelationship between the Fredholm determinant and the Ruelle zeta function[26] which is defined as

ζ (z, q) = ζϕ (z, q) = exp

( ∞∑n=1

zn

nZn (q)

).

We have then ζ(z, q) = 1D0(z,q)

. If the charts are linear we obtain an expressionof the partition function as the difference of Tr(Ln

q) and a constant by Tr(Lnq),

like in (17) for the Kac-model. So that in this case are also related thedeterminant and zeta. For d ≥ 2 the connection comes from (20).

Another result about the transfer operators Lq is the relationship betweenthe spectral radius ρ(Lq) and the topological pressure, which is ρ(Lq) =exp(P(qϕ)). This was proved by Ruelle for the operators (14) and for operatorsimilar to (17) in the above quoted references. To obtain an expressionin terms of the free energy T(q) we just consider renormalized operatorsexp(−qP(ϕ))Lq and so the leading eigenvalue results exp(T(q)). For simplicity,we also denote the renormalized operators by Lq. In [21, 22] it was establishedthe analyticity of the map q −→ ρ(Lq), provided conditions in the style of(C1)–(C3) were fulfilled, and consequently the absence of phase transitions.

The following proposition will be useful to obtain a description of thetransfer operators spectrum.

Proposition 2 The spectrum of the operators L = φCψ , where Cψ is the compo-sition operator Cψ (χ) (z) = (χ ◦ ψ) (z), acting on the space of functions A∞ (U)

is discrete. It consists in eigenvalues En = {φ (z) (Dψ (z))n} where z is a f ixed

point of ψ together with 0 as unique accumulation point.

Proof The fact that the operators L = φCψ have discrete spectrum is actuallydue to [17]. Let ψ ∈ A∞ (D) , we have the eigenvalue equation Lχ (z) =φ (z) χ (ψ (z)) = Eχ (z) . Clearly if χ (z) �= 0 then an eigenvalue of L is E =


φ (z) , where z is a fixed point of ψ. If χ (z) = 0 then differentiating, withrespect to z, the above eigenvalue equation is obtained.

Dφ (z) × χ (z) + φ (z) × Dχ (z) Dψ (z) = EDψ (z) .

Thus if Dφ (z) �= 0 then E = φ (z) Dψ (z) . Now the set of eigenvalues of L(recall that the spectrum is discrete) is

En = {φ (z) (Dψ (z))n} .

Recall that by the Earle-Hamilton theorem ‖Dψ (z)‖ < 1, therefore 0 is theonly point of accumulation .

Notice that Tr (L) =∞∑

n=1En =

∞∑n=1

φ (z) (Dψ (z))n = φ(z)

det(1−Dψ(z)), the Mayer

trace formula. ��

Remark The above result describes indeed the spectrum of the transferoperators since they are finite sums of composite ones.

Now we shall show that the Ruelle zeta function determines the equilibriumstate for a broader class of potentials than in [25].

Proposition 3 Let f : X → X be an expansive homeomorphism in a compactmetric space with the specif ication property and let ϕ1, ϕ2 ∈ ν f (X). Under theseconditions holds ζϕ1 (z, q) = ζϕ2 (z, q) =⇒ Sϕ1 = Sϕ2 (Sϕ1 ,Sϕ2 are the unmarkedorbit spectra of the potentials ϕ1, ϕ2 as def ined at the end of Section 2).

Proof We have ζϕ (z,q) = exp(∞∑

n=1

zn

n Zn(q)), with Zn(q) = ∑x∈Pn( f)

exp(Sn(qϕ)(x)).

The power expansion determines an analytical function in the disc |z| <

exp (T (q)) exp (−qP (ϕ)) . If we have an expression of the form B (q) =N∑

i=1λ

qi ,

λi > 0, then from Newton identities we deduce that B (q) uniquely determinesthe λi, it just needs to know B (1) , B (2) ,..., B (N) . This can be appliedto the finite sum

∑x∈Pn( f )

[exp (Sn (ϕ (x)))

]q and so Zn (q) uniquely determines

the terms Sn (ϕ (x)) , in turn the coefficients Zn (q) are recovered from theexpansion by differentiation with respect to q. In this way the spectrum Sϕ isuniquely determined from the zeta function. ��

Definition A matrix H = (ai, j

)is typical when the numbers log ai, j are

rationally independent, or equivalently if no non trivial product of powers ofthe aij′s (with integer exponents) is equal to 1.


Now we state the main result of this section:

Theorem 3 For spin lattice systems and potentials for which conditions

(C1)–(C3) are fulf illed, let H = (ai, j) be a N × N matrix withN∏

i=1Ei =∑

σ∈Pn

a1,σ (1)...aN,σ (N), where Ei = Ei (q) are the eigenvalues of the transfer op-

erator Lq = Lqϕ (see (18) for the def inition). If H is a typical matrix, then thephenomenon of multifractal rigidity is verif ied, i.e., the multifractal spectrumEϕ (α) (c.f. (7)) determines the spectrum Sϕ.

Proof The scheme of proof is as follows. Firstly it is naturally establishedthat the multifractal spectrum of local entropies determines the free energyTϕ (q), since it is the Legendre transform of the spectrum map Eϕ (α) . Thenwe consider the Fredholm determinant D (z, q) and the map β (q) = 1

ρ(Lq)=

exp (−T (q)) , so that D (β (q) , q) = 0. Let P (z) be an analytic map such thatP (β (q)) = 0 and with β (q) determining P. We show that P (z) is a factorof D (z, q), but we also will prove that it is not possible to write D (z, q) =P (z, q) Q (z, q), where P, Q are non-constant maps. So that the Fredholmdeterminant is in some sense “minimal”, and then β (q) determines the Fred-holm determinant. By the relationship of D (z, q) with the zeta function andby Proposition 3, the claim of the theorem will be proved.

For the above procedure we use an approach based on Tuncel developments

which combines algebraic and dynamic technics [31]. Let R = {k∑

i=0nia

qi : ni ∈

Z, ai > 0}, if we set exp = {aq : a ∈ R+} then Z

[exp

] = R, i.e. R is the ring ofintegral combinations of elements in exp, or we can write

R = {β : R → R :β (q) =k∑

i=0nia

qi }. If the potential ϕ depends on a finite

number of coordinates, for instance ϕ = ϕ(xi, x j

), then it can be defined

a family of matrices H (q) with coefficients in R = Z[exp

]by H (q) ={

0 if Ai, j = 0expq ϕ (x) if Ai, j = 1 , with x0 = i, x1 = j. If β (q) = βA (q) = ρ (A (q)),

it is proved in [31] that β (q)is analytic and βA (1) = log E1, where E1 is theleading eigenvalue of A = A (1) , existing by the Perron–Frobenius theorem.

In our case with a potential which in general depends on the wholeconfiguration we shall take β (q) = 1

ρ(Lq)= exp (−T (q)) , which as we point

out was proved to be analytic and verifies D (β (q) , q) = 0. Recall that byProposition 2 the transfer operators have discrete spectrum and so we can

put D (z, q) = det(1 − zLq

) =∞∏

n=1(1 − zEn (q)) , where E1 (q) = exp (T (q)) ,

so that the z−zeros of the Fredholm determinant are the inverses of theeigenvalues of Lq.

As we anticipate at the beginning of the proof we consider a map P (z, q)

with P (β (q) , q) = 0, analytic in z and whose expansion has coefficients


in R. Let F be the field of fractions of R and let G be the set of ex-pansions of analytic maps with coefficients in F . We consider an idealI in G given by F ∈ I if and only if F can be expressed as F = Q/Rwhere Q = Q (z, q)is an analytic map in z with expansion with coefficientsin R and Q (β (q) , q) = 0 for some analytic function β (q) and R ∈ R.

By the analyticity of β (q) the choice does not depend on R. So I ={F : F can be expanded with coefficients in F , and F (β (q) , q) = 0

}. Let I =

PG for some P with coefficients in F , we shall show that the expansion hasactually coefficients in R. We have that the Fredholm determinant belongs toI and so it can be written: D (z, q) = P (z, q) Q (z, q) , where P and Q havecoefficients in F and D has expansion in R. We then have

D =∞∑

n=0

anzn, with an =∑in∈In

Min Aqin ∈ R, In finite,

P =∞∑

n=0

b nzn, with b n =

∑jn∈Jn

N jn Bqin

∑jn∈Jn

N′jn B

′qin

∈ F, Jn finite,

Q =∞∑

n=0

cnzn, with cn =

∑�n∈Ln

U�n Cq�n∑

�n∈Ln

U ′in C

′qin

∈ F, Ln finite.

For any positive integer n let Sn be the subgroup of R+ generated byAin , B jn B

′jn , C�n , C

′in and Z

[Sn

]is a unique factorization domain. We have

a0 + a1z + ... + anzn = (b 0 + b 1z + ... + arzr)(c0 + c1z + ... + cn−rzn−r

), then

each bi can be expressed as bi = b i/b with b i ∈ Z[Sn

]as well as any

ci = ci/c with ci ∈ Z[Sn

]and for some b , c such that

(b , b 1, ..., b r

) =1, (c, c1, ..., cn−r) = 1. Hence the following expression is an equation inZ[Sn

]bc (a0 + a1z + ... + anzn) = (

c0 + c1z + ... + cn−rzn−r) (

b 0 + b 1z + ... +b rzr

), since Z

[Sn

]is a unique factorization domain each factor of bc must

divide all the b i or all the ci, and besides is invertible. Thus c is a “monomial”and so P has actually coefficients in R. Therefore if P (z, q) has coefficientsin R and β (q) is a z−zero of P then this map is a factor of the Fredholmdeterminant D (z, q) .

Next we prove the “minimality” of the Fredholm determinant, we

consider a “truncation” DN (z, q) :=N∏

n=1(1 − zEn (q)) ∈ R [z]. In this

way DN (z, q) = 1 + (∑

iEi)z + (

∑i, j

Ei E j)z2 + ... + [(−1)n ∏i

Ei]zN . Another

expression for the Fredholm determinant is D (z, q) = 1 +∞∑

n=1

Dn (q) zn,


where Dn (q) = ∑(i1,...,im)

i1+...+im=n

(−1)m

m!m∏

j=1

1i j

Tr(Li jq ), so that DN(z, q) = 1 + Tr(Lq)z +

Tr(L2q)z + ... +

[ ∑(i1,...,im)

i1+...+im=n

(−1)m

m!m∏

j=1

1i j

Tr(Li jq )]zN .

Let us assume that D (z, q) = P (z, q) Q (z, q) , as we have seenP, Q have expansions with coefficients in R if D (z, q) does. Wecompare the coefficients in each N−truncation of D and P.Q. ThusDN (z, q) = 1 + (

∑i

Ei)z + (∑i, j

Ei E j)z2 + ... + [(−1)n ∏i

Ei]zN = [ ∑j0∈J0

N j0 Bqi0 +

(∑

j1∈J1

N j1 Bqi1)z + ... + (

∑jr∈Jr

N jr Bqir )z

r] × [ ∑�0∈L0

U�0 Cq�0

+ (∑

�1∈L1

U�1 Cq�n1

)z + ... +(

∑�N−r∈LN−r

U�N−r Cq�N−r

)zN−r].Notice that the product of the eigenvalues Ei, i = 1, ..., N can be con-

sidered as the determinant of certain N × N−matrix H = (ai, j), soN∏

i=1Ei =∑

σ∈Pn

a1,σ (1)...aN,σ (N), where Ei = Ei(q), ai, j = ai, j(q) and Pn is the group of per-

mutations of n−elements. BesidesN∑

i=1Ei = Tr(H) = ∑

iai,i. Since H is typical:

an1i1, j1 an2

i2, j2 , ..., ankik, jk �= 1 for any (i1, i2, ..., ik) ( j1, j2, ..., jk)

and (n1, n2, ..., nk) ∈ Zk. (21)

The coefficient of zr in the expansion of D (z, q) is of the forma1,σ (1)...aN,σ (N)

ai1,i1 ...air,ir, (22)

where σ ∈ Pn fixes (i1, ..., ir), and the coefficient of zN−r is of the forma1,σ (1)...aN,σ (N)

ai1,i1 ...aiN−r,iN−r

(23)

with σ ∈ Pn fixing (i1, ..., iN−r) .

Then, we have∑

σ∈Pn

a1,σ (1)...aN,σ (N) = ∑jr,�N−r

N jr U�N−r Bqjr C

q�N−r

, so that there

is a correspondence between a1,σ (1)...aN,σ (N) and the coefficients Bqir C

q�N−r

.

Thus comparing the coefficients of zr we have Bqjr C

q�0

= a1,σ (1)...aN,σ (N)

ai1 ,i1 ...air ,irand

also a similar expression for zN−r. If σ ∈ Pn does not have fixed pointsthen a1,σ (1)...aN,σ (N) appears in the constant term of the development of theD (z, q) , but it is not possible to write it as a product of the coefficientsBq

jr Cq�N−r

. To illustrate this, consider the cyclic permutation σ = (1, 2, 3) andthe sum

∑σ∈P3

a1,σ (1)a2,σ (2)a3,σ (3), which of course includes σ . The coefficient of

z2 is a sum of terms ai. ja j,i and ai.ia j, j. Now a1,2a2,3a3,4 must be of the formai. ja j,iam,n, which could not be possible by (21). ��


As we have pointed out Pollicott and Weiss provided examples of locallyconstant potentials in which the rigidity phenomenon is not verified. Morespecifically they found finite range potentials ϕ1 ,ϕ2 with the same free energybut non-equivalent in the sense that ϕ1 is not cohomologous to ϕ2 ◦ τ , whereτ is a homeomorphism which commutes with the Bernoulli shift. To ensurethe rigidity is imposed the condition of genericity (see [25] for the definition)on the matrix Lϕ (defined at the beginning of this section) associated to thepotential ϕ.

When the transfer operator Lq is restricted to the set of locally constantpotentials it is reduced to a matrix Lϕ . Now if the genericity condition isimposed on the matrix H, originated by the truncation of the Fredholm deter-minant, then DN (z, q) determines the matrix and the potential. The genericitycondition allows to recover the coefficients of Dn (q) in the expansion ofDN (z, q).

Conclusion The local multifractal rigidity was proved on weaker conditionsthan those of [25], say expansiveness and specification for the dynamicsand potentials belonging the bounded distortion class, instead of the Höldercontinuous maps which are included in our wider class. On the other handwas proved a rigidity phenomenon for long range potentials, so extending theresults of [25], valid for generic finite range potentials.

5 A Case with Infinite Alphabet

We consider now a lattice system with countable spins, i.e. a system modelledby a Markov subshift �+

A = {x = (xi)i∈N : xi ∈ I, ∀i ∈ N, Axi,xi+1 = 1

}, I infinite

countable. Let f : [0, 1] → [0, 1] be the Gauss map, i.e f (t) = 1t mod 1. If

any t ∈ [0, 1] is represented by its continued fraction t = 1i1+ 1

i2+ 1i3+ ...

, then the

assignation t � ι = (in)n∈N gives a symbolic representation of the dynamicalsystem (I, f ). More generally if f is an analytic expanding map a symbolicrepresentation is obtained via Markov partitions. We consider the potentialϕ (t) = log

∣∣ f ′ (t)∣∣ and the spin system induced the Gauss map in the way

described above.For every positive integer n holds: f n (t) = t if and only if in = in+k, for every

k, where ι = (in) is the infinite sequence associated to t. Hence the followingnotation can be introduced: for any number t with period n, with respect to f,the associated sequence will be denoted by [i1...in] .

The partition function for the system (I, f ) reads:

Zn (q) =∑

x∈Pn( f )

n−1∏j=0

∣∣∣∣( f′) j

(t)

∣∣∣∣q

, (24)


setting φq := exp (qϕ) and replacing t by its symbolic representation from thecontinued fraction, the partition function for the associated spin system can beexpressed as:

Zn (q) =∑

(i1,...,in)

n−1∏j=0

φq([

i1+k...in, i1, .., ik])

. (25)

the convergence, i.e. the existence of the “thermodynamic limit”, is ensuredif∣∣φq (t)

∣∣ ∼ |t|γ , as t → 0, for some γ = γ (q) > 1. This condition is satisfiedwith γ = 2q.

To define the transfer operators let us consider the Markov partition P ={In = [ 1

n+1 , 1n )}n∈N, we have f |In (t) = 1

t − n, so f |In is analytic if t �= 0. and|( f 2)′| ≥ 4. For the special case where we have a Markov partition P = {In} forexpanding analytic maps the charts ψn can be defined as the branches of f |In,

in our case it is ψn (t) = 1t+n , being φq ◦ ψn analytic in a complex neighborhood

of each In. Now the transfer operator becomes:

Lq (κ) (z) =∞∑

n=1

(1

z + n

)2q

χ

(1

z + n

), (26)

where q must be > 12 by convergence reasons already mentioned. These

operators are proved to be nuclear in some adequate functional space, indeedfor this can be taken A∞ (U) such that ψn(U) ⊂ U and φq ◦ ψn ∈ A∞ (U). Theopen complex set U can be choose as the disc |z − 1| < 3

2 .

Therefore in this particular case and for the temperature parameter q > 12

the results about multifractal rigidity valid for the finite alphabet case can beextended to infinite spin systems following the scheme of the earlier section.

More general cases are found by considering the so called boundary hy-perbolic maps, which are functions originated by the action of Kleinian finitelygenerated groups � on the hyperbolic disc H2 such that to any point in the limitset � of this action can be assigned a sequence in the generators of the group.These maps f : � → � were introduced, to codify hyperbolic geodesics, bySeries [27, 28] who proved that the system (�, f ) has a Markov partitionwhich leads to symbolic representation by a subshift with an alphabet whichin general does not agree with the generator set of the group. The alphabetis infinite if and only if � contains parabolic elements. The Gauss map is aspecial case of such a map, corresponding to the action of � = SL2 (Z) . Forthe connection of the boundary hyperbolic maps with multifractal analysis onecan see [18, 23].


Appendix

Topological Entropy for Non-Compact Nor Invariant Sets

Let f : X → X be a continuous map and (X, d) a compact metrisablespace. Let U = {U1, ..., UN} be a finite covering of X . A string is definedas a sequence L = (�0, �1, ..., �n−1) such that

{U�0, U�1, ..., U�n−1

} ⊂ U, �i ∈{1, 2, ..., N}. The length of the string L = (�0, �1, ..., �n−1) is denoted by n (L) =n. Let call Wn (U) the set of the strings L with length n for the covering U .

Let

X (L) = U�0 ∩ f −1 (U�1

) ∩ ... ∩ f −n+1 (U�n−1

),

if Z ⊂ X we say that � = {L = (�0, �1, ..., �n−1)} covers Z if

Z ⊂⋃L∈�

X (L) .

For any real number s:

M (Z ,U, s, n) = inf�

∑L∈�

exp (−sn (L)) ,

where the infimum is taken over all collections of strings � ⊂ Wn (U) whichcover Z .

Let

M (U, Z , s) = limn→∞ M (U, Z , s, n) .

There is a unique number s such that M (U, Z , .) jumps from +∞ to 0, now let

htop ( f, Z ,U) = s = sup {s : M (U, Z , s) = +∞} = inf {s : M (U, Z , s) = 0} .

(27)Finally the number

htop ( f, Z ) = lim�(U)→0

htop ( f, Z ,U) , � (U) = diameter of U (28)

is the topological entropy of f restricted to Z .

Gibbs Measures in Lattice Spin Systems

We present here a formulation of the notion of Gibbs states in lat-tice spin models: Let X be “the configuration space” which, as wealready pointed out, is mathematically described as the set �A ={

x = (xi)i∈Z : xi ∈ �, ∀i ∈ Z, Axi,xi+1 = 1}, where A is a k × k matrix with 0, 1

entries and � = {0, 1, 2, ..., k − 1} . The integers i are called the sites and thecorresponding coordinate xi the spin at the site i. The matrix A indicates


which configurations are allowed. From A is defined a probability vectorp = (p0, p1, ..., pk−1) and a stochastic matrix E = (

Ei, j)

i, j=0,...,k−1 (∑

jEi, j = 1)

such that∑

ipi Ei, j = pj. (see e.g. [32]). This space is equipped with the σ -

álgebra B generated by the semi-algebra of elementary cylinders: C−m,...,mα−m,...,αm

={x ∈ �A : xi = αi, i = −m, ..., m}. The Gibbs states will be probability mea-sures defined on (�A,B). In this space are considered as dynamics the shiftσ : �A → �A, σ x = x

′, where x

′i = xi+1. A Gibbs state in the space of symbolic

dynamics is the product measure defined on cylinders by

μ(C−m,...,m

α−m,...,αm

) = pα−m Eα−m,α−m+1 ...Eαm−1,αm . (29)

For a potential ϕ ∈ C (�A) , which physically can be interpreted as a de-scription of the interaction energy between one spin and the remaining, the

statistical sum Sn (ϕ) (x) =n−1∑i=0

ϕ(σ i (x)

)can be decomposed as Sn (ϕ) (x) =

H (x0, x1, ..., xn−1) + W (x0, x1, ..., xn−1 | xn, xn+1, ...) [26], where H describesthe energy of the spins x0, x1, ..., xn−1 and W the interaction of x0, x1, ..., xn−1and the spins xn, xn+1, .... For a configuration x, let us denote by x(n) anymember of �A with x(n)

i = xi, this is the election of a boundary condi-tion for the system. The partition function is defined now as: Zn (ϕ) =∑i0,...,in−1∈�

exp(Sn (ϕ)

(x(n)

)). Finally Gibbs states are defined as measures which

satisfy the equation:

dμ(x0,...,xn−1

) = Zn (ϕ)−1 exp(Sn (ϕ)

(x(n)

)), (30)

for any configuration x and every n ∈ N.

The parallelism between shift dynamical systems and statistical mechanicsof spin systems by interpreting the symbolic sequences as spin configurationsover the lattice Z was done by Sinai [29]. The above analysis is rooted in hisideas.

The definition of Gibbs states used in a “probabilistic context” is usuallygiven as follows: let � be a finite subset of Z and let us denote x� = (xi)i∈� ,

the projection of the sequence x on �. For prescribed conditional probabilitiesP (x� | x�C) let H� (x) be the Hamiltonian describing the energy excess of xover the energy of x�C , which will be done by

P (x� | x�C) = 1Z�

exp (−H� (x)) , (31)

where Z� is the partition function. Here the inverse of the temperature β

is summed into H�, or alternatively can be taken units in such a way thatβ = 1. Thus a probability measure μ is a Gibbs state for a family of conditionalprobabilities P (x� | x�C)� finite ⊂Z if μ

(x� occurs in � | x�C occurs in �C

) =P (x� | x�C), for every x μ-a.e.

To compare this definition with the earlier one notice these analogies: theHamiltionian H� has its correlate in the statistical sum for the potential ϕ,


the finite set � indicates the sites for a finite set of spins and �C the remain-ing. The partition function Z� in (31) is obtained by summing over all theconfigurations x

′which agree with x in �C, while in the definition as function

of the potential ϕ the summation is over the sites whose spins agree with aconfiguration x. This establishes a correlation between both expressions, thesummation indexes in Z� correspond to the boundary conditions in Zn (ϕ) .

Acknowledgements Support of this work by Consejo Nacional de Investigaciones Científicas yTécnicas (CONICET), Universidad Nacional de La Plata and Agencia Nacional de PromociónCientífica y Tecnológica of Argentina is greatly appreciated. F.V. is a member of CONICET.

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Math Phys Anal Geom (2011) 14:321–341DOI 10.1007/s11040-011-9099-x

The Gross–Pitaevskii Functional with a RandomBackground Potential and Condensationin the Single Particle Ground State

Frédéric Klopp · Bernd Metzger

Received: 12 January 2011 / Accepted: 21 September 2011 / Published online: 15 October 2011© Springer Science+Business Media B.V. 2011

Abstract For discrete and continuum Gross–Pitaevskii energy functionals witha random background potential, we study the Gross–Pitaevskii ground state.We characterize a regime of interaction coupling when the Gross–Pitaevskiiground state and the ground state of the random background Hamiltonianasymptotically coincide.

Keywords Random Schrödinger operators · Gross–Pitaevski functional

Mathematics Subject Classifications (2010) 47B80 · 47H40 · 60H25 · 82B20 ·82B44

1 Introduction

The purpose of the present paper is to study some aspects of condensation inthe ground state of the Gross–Pitaevskii energy functional with a disorderedbackground potential. Before discussing our main result, Theorem 3 below,and its physical background, let us first introduce the necessary mathematicalframework (for further information about the Anderson model see e.g. [14, 27,31] and references therein). As they can be treated in a very similar way, weconsider the discrete and the continuum setting simultaneously.

The authors were partially supported by the grant ANR-08-BLAN-0261-01.

F. Klopp (B)Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie,Case 186, 4 place Jussieu F-75252 Paris cedex 05, Francee-mail: [email protected]

B. MetzgerWeierstrass Institute Berlin, Mohrenstr. 39, 10117 Berlin, Germanye-mail: [email protected]

322 F. Klopp, B. Metzger

The continuum setting In Rd, consider the cube �L = [−L, L]d of side

length 2L and volume |�L| = (2L)d. In HL := L2(�L), on the domain DL :=H2(�L), consider HP

ω,L = (−� + Vω)P�L

the self-adjoint Anderson model on�L with periodic boundary conditions. We assume

• � = ∑dj=1 ∂2

j is the continuum Laplace operator;• Vω is an ergodic random potential i.e. an ergodic random field over R

d thatsatisfies

∀α ∈ Nd, ‖‖∂αVω‖x,∞‖ω,∞ < +∞

where ‖ · ‖x,∞ (resp. ‖ · ‖ω,∞) denotes the supremum norm in x (resp. ω).

These assumptions are for example satisfied by a continuum Anderson modelwith a smooth compactly supported single site potential i.e. if

Vω(x) =∑

γ∈Zd

ωγ u(x − γ )

where u ∈ C∞0 (Rd) and (ωγ )γ∈�L are bounded, non negative identically distrib-

uted random variables.

The discrete setting Consider �L = [−L, L]d ∩ Zd ⊂ Z

d the discrete cube ofside length 2L + 1 and volume |�L| = (2L + 1)d. Let HP

ω,L = (−� + Vω)P�L

bethe discrete Anderson model on DL = HL := �2(�L) with periodic boundaryconditions, that is

• (−�)P�L

is the discrete Laplacian;• Vω is a potential i.e. a diagonal matrix entries of which are are given by

bounded non negative random variables, say ω = (ωγ )γ∈�L .

For the sake of definiteness, we assume that the infimum of the (almost sure)spectrum of Hω be 0. We define

Definition 1 (Gross–Pitaevskii Energy Functional [GPEF]) The (one-particle) Gross–Pitaevskii energy functional on the cube �L (in the discreteor in the continuum) is defined by

EGPω,L [ϕ] = ⟨

HPω,Lϕ, ϕ

⟩ + U‖ϕ‖44 (1)

for ϕ ∈ DL and a positive coupling constant U.

For applications, it is natural that this coupling constant be related to |�L|.We refer to the discussion following Theorem 3 for details. One proves

Proposition 2 For any ω ∈ and L � 1, there exists a ground state ϕGP i.e. avector ϕGP ∈ DL such that ‖ϕGP‖2 = 1 minimizing the Gross–Pitaevskii energyfunctional, i.e.

EGPω,L = EGP

ω,L

[ϕGP] = min

ϕ∈DL‖ϕ‖2=1

EGPω,L[ϕ]. (2)

The Gross–Pitaevskii Functional with a Random Background Potential 323

The ground state ϕGP can be chosen positive; it is unique up to a change of phase.EGP

ω,L denotes the ground state energy of the discrete Gross–Pitaevskii functional.

The proof in the continuum case is given in [24]; the proof in the discretecase is similar.

Let HNω,L and HD

ω,L respectively denote the Neumann and Dirichlet restric-tions of Hω to �L. Our main assumptions on the random model are:

(H0) Decorrelation Estimate The model satisfies a finite range decorrela-tion estimate i.e. there exists R > 0 such that, for any J ∈ N

∗ and any sets(Dj)1� j�J , if

infj= j′

dist(Dj, Dj′

)� R,

then the restrictions of Vω to the domains Dj, i.e. the functions (Vω|D j)1� j�J ,are independent random fields.

(H1) Wegner Estimate There exists C > 0 such that, for any compact intervalI and • ∈ {P, N, D},

E[tr(1I(H•

ω,L

))]� C|I|Ld;

(H2) Minami Estimate There exists C > 0 such that, for I a compact intervaland • ∈ {P, N, D},

P[{H•

ω,L has at least two eigenvalues in I}] � C(|I|Ld)2 ;

(H3) Lifshitz Type Estimate Near Energy 0 There exist constants C > c >

0 such that, for L � 1 and any parallelepiped PL = I1 × · · · × Id where theintervals (I j)1� j�d satisfy L/2 � |I j| � 2L, one has

ce−Ld/c � P[{HD

ω|PLhas at least one eigenvalue in [0, L−2]}] ,

P[{HN

ω|PLhas at least one eigenvalue in [0, L−2]}] � Ce−Ld/C

where HDω|PL

(resp. HDω|PL

) is the Dirichlet (resp. Neumann restriction) of Hω

to PL.Let us now discuss the validity of these assumptions.The decorrelation assumption (H0) is satisfied for the discrete Anderson

model described above if the random variables (ωγ )γ∈Zd are i.i.d. (H0) clearlyallows some correlation between the random variables. For the Andersonmodel in the continuum setting, it is satisfied if the single site potential hascompact support and the random variables are i.i.d.

Under the assumption that the random variables are i.i.d and that theirdistribution is regular, it is well known that the Wegner estimate (H1) holdsat all energies for both the discrete and continuum Anderson model (seee.g. [7, 16, 34]).


The Minami estimate (H2) is known to hold at all energies under similarregularity assumptions for the discrete Anderson model (see e.g. [3, 5, 12, 26])and for the continuum Anderson model in the localization regime under morespecific assumptions on the single site potential (see e.g. [6]).

Finally, the Lifshitz tails estimate (H3) is known to hold for both thecontinuum and discrete Anderson model under the sole assumption that thei.i.d. random variables be non degenerate, non negative and 0 is in theiressential range (see e.g. [14, 15, 17]). Though the Lifshitz tails estimate isusually not stated for parallelepipeds but for cubes, the proof for cubes appliesdirectly to parallelepipeds satisfying the condition stated in (H3).

The main result of the present paper is

Theorem 3 (Condensation in the Single Particle Ground State). Assume as-sumptions (H0)–(H3) hold. Denote by ϕ0 the single particle ground state ofHP

ω,L (chosen to be positive for the sake of def initeness) and by ϕGP the Gross–Pitaevskii ground state.

If for L large, one assumes that

U = U(L) = o(

L−d

(1 + (log L)d−2/d+ε) fd(log L)

)

where

fd(ξ) =

⎧⎪⎨

⎪⎩

ξ−1/4 if d � 3,

ξ−1/d log ξ if d = 4,

ξ−1/d if d � 5.

(3)

and ε = 0 in the discrete setting, resp. ε > 0 arbitrary in the continuum case,then, there exists 0 < η(L) → 0 when L → +∞ such that

P[∣∣〈ϕ0, ϕ

GP〉 − 1∣∣ � η(L)}] →

L→+∞0. (4)

The proof of Theorem 3 also yields information on the size of η(L) and onthe probability estimated in (4). Note that the assumption (H1)–(H3) can berelaxed at the expense of changing the admissible size for U .

To appreciate Theorem 3 maybe some comments about the physicalbackground of the Gross–Pitaevskii model, its relationship to Bose–Einsteincondensation and to known results are of interest. Motivated by recentexperiments with weakly interacting Bose gases in optical lattices (see forexample [4]) the fundamental objects of interest are the ground state densityand energy, i.e.

EQM := min�∈ N⊗

sL2(�L)

‖�‖=1

⟨

�,

⎡

⎣N∑

i=1

{−�i + V(xi)} +∑

1�i< j�N

v(∣∣xi − x j

∣∣)⎤

⎦�

⟩

. (5)


The optical lattice is modeled by the background potential V as shown inFig. 1. Assuming a weak interaction limit of the interaction potential v(x, y),the continuum N-particle Gross–Pitaevskii energy functional

EGP = minϕ∈L2(�L)‖ϕ‖2=1

∫

�L

(N∣∣∇ϕ(x)|2 + NV|ϕ(x)

∣∣2 + 4N2πμa|ϕ(x)|4

)dx, (6)

is a mean field approximation of the ground state energy (5), e.g. in threedimensions one has

limN→∞

EQM

EGP= 1.

(see for example [24]) The discrete Gross–Pitaevskii model is then a tightbinding approximation of the continuum one-particle Gross–Pitaevskii func-tional [29, 30]

EGP[ϕ] =∫

R3

(|∇ϕ(x)|2 + V |ϕ(x)|2 + 4Nπμa |ϕ(x)|4)dx.

Another way to derive the discrete Gross–Pitaevskii model starts with adiscretization of (5) yielding the standard description of optical lattices usingthe Bose–Hubbard–Hamiltonian

H = −∑

|n−n′ |=1

c†ncn′ +

∑

n

(σ Vn − μ) nn + 12

U∑

n2n

where c†n, cn are bosonic creation and annihilation operators and nn gives the

particle number at site n (see the survey article [4] and references therein).A mean field approximation then yields the discrete Gross–Pitaevskii energyfunctional [22].

One motivation to study Bose gases is Bose–Einstein condensation (BEC),i.e. the phenomena that a single particle level has a macroscopic occupation ( anon-zero density in the thermodynamic limit) [24]. BEC was introduced in [9]for an ideal Bose gas. Due to naturally arising interactions it took some timeto realize Bose-Einstein condensation experimentally [8, 19].

Fig. 1 An example of abackground potentialmodeling an optical lattice[28]


As we will see, also the formal description is more elaborated. To motivatethe definition of BEC for an interacting Bose gas at vanishing temperaturewe follow the continuum approach in [24]. To formalize the concept of amacroscopic occupation of a single particle state we recall the definition ofthe one-particle density matrix [24], i.e. the operator on L2(R3) given by thekernel

γ(x, x′) = N

∫

�QM (x, x2, . . . , xN) �QM (x′, x2, . . . , xN

) N∏

j=2

dx j

with the normalized ground state wave function �QM of the many BosonHamiltonian. BEC in the ground state means that the projection operator γ

has an eigenvalue of order N in the thermodynamic limit.For the ideal Bose gas the multi-particle ground state can be represented as

a product

�QM (x1, . . . , xN) = ∏Ni=1 ϕ0 (xi)

of the single particle ground state ϕ0. The one-particle density matrix thenbecomes

γ(x, x′) = N ϕ0(x)ϕ0

(x′) .

Thus the definition of BEC above is natural and can also be related to thethermodynamic formalism (see e.g. [23, 24] and references). In particular, it isof interest to consider BEC for the ideal Bose gas with a random backgroundpotential. In this case the Lifshitz tail behavior at the bottom of the spectrummakes a generalized form of Bose–Einstein condensation possible even ford = 1, 2 (see [23] and references cited there).

The situation in the Gross–Pitaevskii-limit is close to the situation for theideal Bose gas [24]. The one-particle density matrix is asymptotically given by

γ(x, x′) N→∞∼ N ϕGP(x)ϕGP (x′) . (7)

Physically the content of (7) is that all Bose particles will condensate in the GPground state motivating the definition of complete (or 100%) BEC in [24].

The present publication is a first step toward the analysis of the finestructure of the Gross–Pitaevskii ground state in a random background po-tential. We want to understand how ϕGP is related to the eigenstates of thesingle particle Hamiltonian. This question has been studied in various cases inparticular in the following two settings.

Suppose lim inf|x|→∞ V(x) = ∞ , i.e. the Bosons are trapped by a poten-tial tending to ∞. Then the spectrum of the unrestricted single particleHamiltonian is discrete. Furthermore the distance between the first two eigen-values of HP

ω,L is strictly positive in the asymptotic limit L → ∞. AssumingNa → 0 in the continuum setting, respectively NU → 0 in the context of thediscrete Gross–Pitaevskii model, the interaction energy is a small perturba-tion of the single particle energy functional. In this situation it is natural,


that in the thermodynamic limit ϕGP and the single particle ground state ϕ0coincide.1

A complementary situation is given if the Bosons are confined to a cube�L with |�L| → ∞ but without a background potential. As described in [24]assuming ρ = N/L3 and g = Na/L in the limit N → ∞ one can prove

limN→∞

1N

1L3

∫∫

γ (x, y)dxdy = 1,

i.e. BEC in the normalized single particle ground state ϕ0 = L−d/2χ�L . Asexplained in [24]

g = NaL

= ρa1/L2

is in this context the natural interaction parameter since “in the GP limit theinteraction energy per particle is of the same order of magnitude as the energygap in the box, so that the interaction is still clearly visible”.

As emphasized in the physics literature (see e.g. [4, 25]), new phenomenalike fragmented BEC (Lifshitz glasses) should occur when Bosons are trappedin a random background potential. Our purpose in this publication is moremodest. We want to understand the natural interaction parameter in a randommedia, s.t. the Gross–Pitaevskii ground state is close to the ground state of thesingle particle Hamiltonian as it is suggested by the situation in the ideal Bosegas. As we will see the setting of Bosons trapped in a random potential is notreally comparable to the two situations described above.

Under our assumptions, near 0 which is the almost sure limit of inf(HPω,L),

we are in the localized regime, i.e. one has pure point spectrum and localizedeigenfunctions. In contrast to the situation with vanishing potential the eigen-states close to the bottom of the spectrum are localized in a small part of �L,i.e. the interaction energy will be larger than in the case of the homogeneousBose gas. In the random case, we determine the almost sure behavior ofthe ground state from information on the integrated density of states (seeLemma 5). Under our weak Lifshitz tails assumption (H3), we obtain thatthe ground state energy is of size (log L)−2/d. When L → +∞, the differencebetween the first two eigenvalues will tend to zero; the speed at which thishappens is crucial in our analysis (see Proposition 10). In our case, we estimatethat, with good probability, it must be at least of order L−d. This differenceis much smaller than the one obtained in the homogeneous Bose gas whereit typically is of order L−2. We deem that the estimate L−d for the spacingis not optimal in the present setting. This estimate is the correct one in thebulk of the spectrum; at the edges, the spacings should be larger. It seemsthat getting an optimal estimate requires a much better knowledge of theintegrated density of states or, in other words, much sharper Lifshitz tails typeestimates (see (H3)) and Minami type estimates that take into account the

1M. Lewin (private communication).


fact that we work at the edge of the spectrum (see (H2)). The combinationof the observations above provides the following asymptotic bound for theinteraction parameter: U = o(L−dh−1

d (log L)). As discussed above we do notbelieve this to be optimal.

Let us now briefly outline the structure of our paper. To prove our result weneed two ingredients. We need an upper bound for the interaction term, i.e. wehave to estimate the ‖.‖4- norm of the single particle ground state ϕ0. At thesame time, we need a lower bound of the distance of the first two single particleeigenvalues asymptotically almost surely (a.a.s.) i.e. with a probability tendingto 1 in the thermodynamic limit. Comparing these two estimates we will seethat, under the assumptions of Theorem 3, it is energetically favorable that theGross–Pitaevskii ground state and the single particle ground state coincide.This will be proven at the end of this publication.

To estimate the interaction term we will prove in Lemma 5 that almostsurely in the thermodynamic limit the single particle ground state is flat, i.e.

‖∇ϕ0‖2 L→∞−−−→ 0 a.a.s.

This, then, yields a bound on the interaction term which is the purpose ofProposition 4.

The a.a.s. lower bound of the distance of the first two single particleeigenvalues is a little bit more intricate and uses the Wegner and Minamiestimates; it is related to the methods developed in [11]. In Lemma 12, we firstestimate the probability that the first two eigenstates and also their localizationcenter are close together. If the localization centers are relatively far away, onecan decouple the eigenstates and treat the first two eigenvalues of each other.This is used in Lemma 13.

2 Estimating the Interaction Term

The main result of this section is an upper bound on EGPω,L − EP

0 [ω, L]. Thisquantity is non negative (see (2)) and we prove

Proposition 4 There exists C > 0, such that, for any p ∈ N, one has

P[EGP

ω,L − EP0

[ω, L

]� CU fd(log L)

]� 1 − L−p (8)

where fd is def ined in (3).

By definition, for ϕ0(ω, L) the ground state of HPω,L, one has

EGPω,L � Eω,L [ϕ0(ω, L)] = EP

0

[ω, L

] + U ‖ϕ0(ω, L)‖44 (9)

resp.

EGPω,L − EP

0

[ω, L

]� U‖ϕ0(ω, L)‖4

4.

To prove Proposition 4, resp. control the interaction term, we first estimatethe ground state energy of the random Schrödinger operator and derive in


Corollary 6 an estimate on the “flatness” of its ground state. We start with theDirichlet and Neumann boundary cases.

Lemma 5 Assume (H3) is satisf ied. Let EP0 (ω, L) be the ground state energy of

HPω,L and denote by ϕ0(ω, L) the associated positive normalized ground state.

Then, for any p > 0, there is a constant C > 0 such that, for L suf f iciently large,

P

[C−1 (log L)−2/d � EP

0 (ω, L) � C(log L)−2/d]

� 1 − L−p. (10)

As Vω is non negative and ϕ0(ω, L) normalized, one has ‖∇ϕ0(ω, L)‖2 �EP

0 (ω, L). Hence, Proposition 4 implies the following “flatness” estimate ofthe ground state.

Corollary 6 Under the assumptions of Proposition 4, for any p > 0, there is aconstant C > 0 such that, for L suf f iciently large,

P[‖∇ϕ0(ω, L)‖2 � C(log L)−2/d] � 1 − L−p. (11)

It may be interesting to note that from a Lifshitz tail type estimate (i.e. theannealed estimate), we recover the (approximate) almost sure behavior of theground state energy of HN

ω,L (i.e. the quenched estimate) (see e.g. [33]).We note that Proposition 4 and Corollary 6 also hold if we replace the

periodic ground state and ground state energy by the Neumann or Dirichletones.

Proof of Lemma 5 Fix � � 1. Decompose the interval [−L, L] into intervalsof length comprised between �/2 and 2�. This yields a partition of �L inparallelepipeds i.e.

�L =⋃

1� j�J

P j

such that

• P j = I1j × · · · × Id

j where the intervals (Ikj )1�k�d satisfy �/2 � |I j

k| � 2�• for j = j′, P j ∩ P j′ = ∅,• J, the number of parallelepiped, satisfies 2−d(L/�)d � J � 2d(L/�)d.

In the continuum model, one can take the parallelepiped to be cubes.Denote by ω|�L the restriction of ω to �L. Furthermore, let ωP,L be the

periodic extension of ω|�L to Zd i.e. for β ∈ �L and γ ∈ Z

d, ωP,Lβ+γ L

= ωβ where

L = 2L + 1 in the discrete case and 2L in the continuum one. As HPω is the

periodic restriction of Hω to �L, we know that EP0 [ω, L] = inf σ(HωP,L) where

this last operator is considered as acting on the full space Rd or Z

d (seee.g. [18]).


We can now decompose Rd or Z

d into ∪γ∈Zd ∪Jj=1 (γ L + P j). By Dirichlet–

Neumann bracketing (see e.g. [14, 16]) HωP,L satisfies as an operator onZ

d or Rd

⊕γ∈Zd ⊕Jj=1 HN

ω|(γ L+P j)� HωP,L � ⊕γ∈Zd ⊕J

j=1 HDω|(γ L+P j)

. (12)

Define

E•0

[ω, �, j

] = inf σ(

H•ω|P j

)for • ∈ {N, D} ; (13)

here, the superscripts D and N refer respectively to the Dirichlet andNeumann boundary conditions. As ωP,L is LZ

d-periodic, H•ω|P j

and H•ω|(γ L+P j)

are unitarily equivalent. The bracketing (12) then yields

inf1� j�J

EN0

[ω, �, j

]� EP

0

[ω, L

]� inf

1� j�JED

0

[ω, �, j

].

Labeling every second interval of the partition of [−L, L] used to constructthe partition of �L, we can partition the interval {1, · · · , J} into 2d sets, say(Jl)1�l�2d such that

(1) if l = l′, Jl ∩ Jl′ = ∅,(2) for j ∈ Jl and j′ ∈ Jl such that j = j′, one has dist(Pj, Pj′) � �/2,(3) there exists C > 0 such that for 1 � l � 2d, C−1(L/�)d � #Jl � C(L/�)d.

Assume R is given by (H0). By (2) of the definition of the partition above,for any l � 2R, all the (H•

ω|P j) j∈Jl , resp. all the (E•

0[ω, �, j ]) j∈Jl (for • ∈ {N, D})are independent. Hence, using (13), we compute

P[EP

0

[ω, L

]> E

]� P

[

infj

ED0

[ω, �, j

]> E

]

�2d∑

l=1

∏

j∈Jl

P[ED

0

[ω, �, j

]> E

]

=2d∑

l=1

∏

j∈Jl

(1 − P

[ED

0

[ω, �, j

]� E

]).

Pick E = c�−2 where c is given by assumption (H3) and

(k log L − c−1 log c)1/d � � � (k log L − c−1 log c)1/d + 1

where k will be chosen below. Applying the Lifshitz estimate (H3), we obtain

P[EP0

[ω, L

]> E] �

2d∑

l=1

(1 − e−k log L/c)#Jl

�2d∑

l=1

exp(−#Jl e−k log L/c) � O(L−∞)

if we choose k < cd as C−1(L/�)d � #Jl � C(L/�)d for 1 � l � 2d.


Hence, for C sufficiently large, we have

P

[EN

0

[ω, L

]� C (log L)−2/d

]� 1 − O(L−∞).

To estimate from below, we use again (13) to get

P[EP

0

[ω, L

]� E

]�∑

j∈JP[EN

0

[ω, �, j

]� E

]).

Pick E = C�−2 where C is given by assumption (H3) and(k log L − C−1 log C

)1/d � � �(k log L − C−1 log C

)1/d + 1

where k will be chosen below. As #J � C(L/�)d, applying the Lifshitz estimate(H3), we obtain

P[EN

0

[ω, L

]� E

]� C

(L�

)d

e−k log L/C � L−p

if we choose k > (d + p)C. Hence, we have, for C sufficiently large,

P[EN

0

[ω, L

]� (log L)−2/d/C

]� 1 − L−p.

This completes the proof of Lemma 5. ��

To prove estimate (8), we will use the spectral decomposition of −�PL.

Though the arguments in the discrete and continuum cases are quite similar, itsimplifies the discussion to distinguish between the discrete and the continuumcase rather than to introduce uniform notations. We start with the discretecase.

Lemma 7 There exists C > 0 such that, for ε ∈ (0, 1) and L ∈ N satisfying L·ε � 1 one has for u ∈ �2(Zd/(2L + 1)Zd) with ‖u‖2 = 1 and 〈−�P

Lu, u〉 � ε2 theestimate

‖u‖4 � Cgd(ε) where gd(ξ) =

⎧⎪⎨

⎪⎩

ξd/4 if d � 3,

ξ | log ξ | if d = 4,

ξ if d � 5.

(14)

Proof The spectral decomposition of −�PL is given by the discrete Fourier

transform that we recall now. Identify �L with the Abelian group Zd/(2L +

1)Zd. For u ∈ HL, set

u = (uγ

)|γ |�L where uγ = 1

(2L + 1)d/2

∑

|β|�L

uβ · e−2iπγβ/(2L+1). (15)

Then, one checks that (see e.g. [20])

(−�PLu

) ˆ = (h (γ ) uγ

)|γ |�L where h(γ ) = 2d − 2

d∑

j=1

cos(

2π γ j

2L + 1

)

. (16)


Pick u ∈ �2(Zd/(2L + 1)Zd) with ‖u‖2 = 1 and 〈−�PLu, u〉 � ε2 and write

u = ∑kε

k=0 uk where kε ∈ N, − log ε � kε < − log ε + 1 and

• u0 = u · 1|γ |<εL• for 1 � k � kε − 1, uk = u · 1ek−1εL�|γ |<ekεL• ukε

= u · 1ekε εL�|γ |

where u denotes the discrete Fourier transform defined in (15). Then, for k =k′, 〈uk, uk′ 〉 = 0 and, using (16), for k � 1,

C−1kε∑

k=0

(ek−1ε

)2 ‖uk‖22 �

kε∑

k=0

⟨−�PLuk, uk

⟩ = ⟨−�PLu, u

⟩� ε2

i.e.kε∑

k=0

e2k ‖uk‖22 � C. (17)

Hence, using (15) and Hölder’s inequality, we compute

∣∣(uk)β

∣∣ = 1

(2L + 1)d/2

∣∣∣∣∣∣

∑

ek−1εL�|γ |<ekεL

(uk)γ

e−2iπγβ/(2L+1)

∣∣∣∣∣∣

� 1(2L + 1)d/2

⎛

⎝∑


|(uk)γ |p

⎞

⎠

1/p ⎛

⎝∑


1

⎞

⎠

1/q

� 1(2L + 1)d/2 ‖uk‖p

(ekε (2L + 1)

)d/q.

So, for p = q = 2, one gets

‖uk‖∞ � ‖uk‖2(ekε

)d/2. (18)

Then, using (17), we compute

‖u‖4 �kε∑

k=0

‖uk‖4 �kε∑

k=0

√‖uk‖2‖uk‖∞ � Ckε∑

k=0

ek(d−4)/4εd/4 � Cgd(ε)

where gd is defined in (14). This completes the proof of Lemma 7. ��

Remark 8 Lemma 7 is essentially optimal as, for L sufficiently large,

• the trial function

uγ ={

ε if γ = 0,

(2L + 1)−d/2 if γ = 0,

satisfies 1 � ‖u‖2 � 1 + ε, 〈−�PLu, u〉 � Cε2 and ‖u‖4 � ε/C;


• the trial function

uγ ={

(2εL + 1)−d/2 if |γ | � εL,

0 if |γ | > εL,

satisfies ‖u‖2 = 1, 〈−�PLu, u〉 � Cε2 and ‖u‖4 � εd/4/C.

We now turn to the continuum case.

Lemma 9 Fix η ∈ (0, 1/4). There exists C > 0 such that, for ε ∈ (0, 1), n > (d −2)η−1 + 1 and L ∈ N satisfying L · ε � 1 one has for u ∈ Hn(Rd/(2L)Zd) with〈−�P

Lu, u〉 � ε2 the norm estimate

‖u‖4 � Cgd,n(ε)‖u‖η

Hn where gd,η(ξ) =

⎧⎪⎨

⎪⎩

ξd/4 if d � 3,

ξ 1−η| log ξ | if d = 4,

ξ 1−η if d � 5.

Proof We now use the Fourier series transform to decompose −�PL. Identify

�L with the Abelian group Rd/2LZ

d. For u ∈ HL, set

u = (uγ

)γ∈Zd where uγ = 1

(2L)d/2

∫

�L

u(θ) · e−π iγ θ/Ldθ. (19)

Then,

u(θ) = 1(2L)d/2

∑

γ∈Zd

uγ eπ iγ θ/L (20)

and(−�P

Lu) ˆ =

(∣∣∣πγ

L

∣∣∣2

uγ

)

|γ |�Lif u ∈ DL. (21)

Pick u as in Lemma 9 and decompose it as in the proof of Lemma 7 i.e. writeu = ∑kε

k=0 uk where kε ∈ N, − log ε � kε < − log ε + 1 and

• u0 = u · 1|γ |<εL• for 1 � k � kε − 1, uk = u · 1ek−1εL�|γ |<ekεL• ukε

= u · 1ekε εL�|γ |

where u denotes the Fourier series transform defined in (19) and (20).The control on uk for 0 � k � kε − 1 is obtained in the same way as in

the proof of Lemma 7 namely the estimate (18) holds for 0 � k � kε − 1.The additional ingredient that we need is to obtain a control over the largefrequency components.

Recall that

‖u‖2Hn =

∑

γ∈Zd

(

1 +∣∣∣πγ

L

∣∣∣2)n/2

|uγ |2


Fix r > d. For notational convenience, write v = ukεand compute

|v(θ)| = 1(2L)d/2

∣∣∣∣∣∣

∑

ekε εL�|γ |

(v)γ

e−iπγ θ/L

∣∣∣∣∣∣

� 1(2L)d/2

∑

ekε εL�|γ |

[∣∣∣πγ

L

∣∣∣r/2 ∣

∣(v)γ∣∣] ∣∣∣πγ

L

∣∣∣−r/2

� 1(2L)d/2

⎛

⎝∑


∣∣∣πγ

L

∣∣∣r |(v)γ |2

⎞

⎠

1/2 ⎛

⎝∑

ek−1εL�|γ |

∣∣∣πγ

L

∣∣∣−r

⎞

⎠

1/2

� C

⎛

⎝∑


∣∣∣πγ

L

∣∣∣r−2/q |(v)γ |2/p ·

∣∣∣πγ

L

∣∣∣2/q |(v)γ |2/q

⎞

⎠

1/2

� C‖v‖1/pHrp−2p/q · ⟨−�P

Lv, v⟩1/(2q)

� C‖v‖η

Hn · ε1−η

if p = η, q = 1 − η and r = (n − 1)η + 2 > d as n > (d − 2)η−1 + 1. One thencompletes the proof of Lemma 9 in the same way as that of Lemma 7. ��

Proof of Proposition 4 In the discrete case Proposition 4 is a consequence ofLemmas 5 and 7 with ε = C(log L)−1/d.

To be able to apply Lemma 9 to ϕ0(ω, L) in the continuum case, we needto show that, for any n > d, ϕ0(ω, L) ∈ Hn with a bounded depending onlyon n not on L or ω. Therefore we use the first assumption on the randomfield Vω i.e. that, for any α ∈ N

d, ‖∂αVω‖ω,x,∞ < +∞. Hence, as ϕ0(ω, L) is aneigenvector of −�P

L + Vω, using the eigenvalue equation

−�PLϕ0(ω, L) = (

EP0 [ω, L] − Vω

)ϕ0(ω, L)

inductively, we see that

‖‖ϕ0(ω, L)‖Hn‖ω,∞ < +∞.

Proposition 4 in the continuum case is then a consequence of Lemmas 5 and8 with ε = C(log L)−1/d. This completes the proof of Proposition 4. ��


3 The Spectral Gap of the Random Hamiltonian

The main result of the present section is

Proposition 10 Let the f irst two eigenvalues of HPω,L be denoted by EP

0 [ω, L] <

EP1 [ω, L]. Then, for p > 0, there exists C > 0 such that, for L suf f iciently large

and η ∈ (0, 1), one has

P[EP

1 [ω, L] − EP0 [ω, L] � ηL−d] � Cη

[1 + (log L)d−2/d+ε

] + L−p

with ε = 0 in the discrete setting resp. ε > 0 arbitrary in the continuum case.

In the localization regime, both the level-spacing and the localization cen-ters spacing have been studied in e.g. [11, 13]. The main difficulty arising in thepresent setting is that the interval over which we need to control the spacingis of length C(log L)−2/d; it is large compared to the length scales dealt within [11, 13].

Our analysis of the spectral gap relies on the description of the groundstate resulting from the analysis of the Anderson model Hω in the localizedregime (see e.g. [14], [31]). Under the assumptions made above on Hω, thereexists I a compact interval containing 0 such that, in I, the assumptions of theAizenman–Molchanov technique (see e.g. [1, 2]) or of the multi-scale analysis(see e.g.[10]) are satisfied. One proves

Lemma 11 ([10, 21]) There exists α > 0 such that, for any p > 0, there existsq > 0 such that, for any L � 1 and ξ ∈ (0, 1), there exists I,δ,L ⊂ such that

• P[I,δ,L] � 1 − L−p,• for ω ∈ I,δ,L, one has that, if ϕn,ω is a normalized eigenvector of Hω|�L

associated to En,ω ∈ I, and xn(ω) ∈ �L is a maximum of x �→ |ϕn,ω(x)| on�L then, for x ∈ �L, one has,

|ϕn,ω(x)| � Lq ·{

e−α|x−xn(ω)| in the discrete case,e−α|x−xn(ω)|ξ in the continuum case.

(22)

Note that, for a given eigenfunction, the maximum of its modulus need notbe unique but two maxima can not be further apart from each other than adistance of order log L. So for each eigenfunction, we can choose a maximumof its modulus that we dub center of localization for this eigenfunction.

To prove Proposition 10, we will distinguish two cases whether the localiza-tion centers associated to EP

0 [ω, L] and EP1 [ω, L], say, respectively x0(ω) and

x1(ω) are close to or far away from each other.In Lemma 12, we show that the centers of localization being close is a very

rare event as a consequence of the Minami estimate.In Lemma 13, we estimate the probability of E0[ω, L] and E1[ω, L] being

close to each other when x0(ω) and x1(ω) are far away from each other. In this


case, E0[ω, L] and E1[ω, L] are essentially independent of each other, and theestimate is obtained using Wegner’s estimate.

Lemma 12 For p > 0, there exists L0 > 0 such that, for λ > 0, L � L0 and η ∈(0, 1), one has

P

[EP

1 [ω, L] − EP0 [ω, L] � η L−d,

|x0(ω) − x1(ω)| � λ(log L)1/ξ

]

� Cη(log L)d/ξ−2/d + L−p

with ξ = 1 in the discrete setting resp. ξ > 1 arbitrary in the continuum case.

Proof Let us start with the discrete setting. Fix p > 0 and let q be given byLemma 11. The basic observation following from Lemma 11 is that, for ω ∈I,δ,L, if xn(ω) is the localization center of ϕn(ω, L) and l � L, then

∥∥(H0

ω − EPn (ω, L)

)ϕn(ω, L, l)

∥∥ + |‖ϕn(ω, L, l)‖ − 1| � CLqe−αl. (23)

where

• H0ω = [HP

ω,L]|xn(ω)+�l is HPω,L restricted to the cube xn(ω) + �l,

• ϕn(ω, L, l) = 1xn(ω)+�l ϕn(ω, L) is the eigenfunction ϕn(ω, L) restricted tothe cube xn(ω) + �l.

To apply the observation above we pick a covering (C j)0� j�J of �L by cubesof side length of order log L i.e. �L ⊂ ⋃

0� j�J C j. Then the number of cubes Jcan be estimated by J � CLd(log L)−d and there exists C > 0 (depending onλ, q and ν) such that, if |x0(ω) − x1(ω)| � λ log L and l � Cλ log L, there existsa cube C j (containing x0(ω)) such that, for L sufficiently large

1∑

k=0

(‖ (H jω − EP

k (ω, L))ϕk(ω, L, j )‖ + |‖ϕk(ω, L, j )‖ − 1|)

+|〈ϕ0(ω, L, j ), ϕ1(ω, L, j )〉| � L−ν/2

where we have set q − Cλα < −ν (see (23)) and

• H jω is the operator Hω restricted to the cube C j + �l,

• ϕk(ω, L, j ) = 1C j+�l ϕk(ω, L) for k ∈ {0, 1}.Let C be given by Lemma 5 and define I = [0, 2C(log L)−2/d]. Decompose

I ⊂ ∪2M+1m=0 Im where

• Im are intervals of length 4ηL−d,• for m ∈ {0, . . . , M − 1}, I2m ∩ I2(m+1) = ∅ = I2m+1 ∩ I2m+3,• for m ∈ {0, . . . , M}, I2m ∩ I2m+1 is of length 2ηL−d.

One can choose M � CLd(log L)−2/dη−1. This implies that, for L sufficientlylarge,

{

ω; EP1 [ω, L] − EP

0 [ω, L] � η L−d

|x0(ω) − x1(ω)| � λ log L

}

⊂ 1 ∪ 2


where 1 = \ I,δ,L and

2 =J⋃

j=1

2M+1⋃

m=0

{(Hω)|C j+�l has two eigenvalues in Im}.

By Lemma 11, we know that

P[1] � L−p

Minami’s estimate (H.2) and the estimate on M tells us that

P[2] � CL2d(log L)−d−2/dη−1 (ηL−d(C log L)d)2 � Cη(log L)d−2/d.

This completes the proof for the discrete setting. The proof for the con-tinuum case is very similar. One has to replace 1xn(ω)+�l by a smooth versionof the characteristic function of the cube xn(ω) + �l (see for example [32]),resp. change the length scale log L to (log L)1/ξ in the side length of the boxeswhere one restricts the eigenfunctions. This is necessary because of the weakerestimate in Lemma 11. This completes the proof of Lemma 12. ��

We now estimate the probability of the spectral gap being small conditionedon the fact that the localization centers are far away from one another. Weprove

Lemma 13 For any p > 0, there exists λ > 0 and C > 0 such that, for Lsuf f iciently large and η ∈ (0, 1), one has

P

[EP

1 [ω, L] − EP0 [ω, L] � η L−d,

|x0(ω) − x1(ω)| � λ(log L)1/ξ

]

� Cη + L−p

with ξ = 1 in the discrete setting, resp. ξ > 1 arbitrary in the continuum case.

Proof Using the same line of reasoning as in the proof of Lemma 12 we givethe proof in the discrete setting.

Fix ν > 2d + p and split the interval [0, C(log L)−2/d] into intervals of lengthL−ν as in the proof of Lemma 12. By Minami’s estimate, we know that, for Lsufficiently large

P[EP

1

[ω, L

] − EP0

[ω, L

]� L−ν

]� C (log L)−2/d Lν L2(d−ν) � L−p. (24)

So we may assume that EP1 [ω, L] − EP

0 [ω, L] � L−ν .As in the proof of Lemma 12, pick a covering of �L by cubes, say (C j)0� j�J

of side length less than log L such that J, the number of cubes, satisfies J �C(L/ log L)d.

Assume that C j is the cube containing x1(ω), E1[ω, L] − E0[ω, L] � η L−d

and |x0(ω) − x1(ω)| � λ log L. Let �cj = �L \ (C j + �3/4λ log L). Define the op-

erators (Hω)|�cj, resp. (Hω)|C j+�λ log L/4 to be the restriction of HP

ω,L to �cj, resp.


C j + �λ log L/4, with Dirichlet boundary conditions. If λ � 8 and L is largeenough, we know that

• dist(x0(ω), ∂�cj) � λ log L − 3/4λ log L − log L � λ log L/8,

• dist(x1(ω), ∂(C j + �λ log L/4)) � λ log L/4• dist(�c

j, C j + �λ log L/4) � λ log L/2 � R

with R > 0 as in the decorrelation assumption (H0). Hence, for λ sufficientlylarge, using the estimate (23) for the operators (Hω)|�c

jand (Hω)|C j+�λ log L/4 , we

know that:

• The operator (Hω)|C j+�λ log L/4 admits an eigenvalue, say E1(ω), that satisfies|E1(ω) − EP

1 (ω)| � L−2ν ;• The operator (Hω)|�c

jadmits an eigenvalue, say E0(ω), that satisfies

|E0(ω) − E0(ω)| � L−2ν . Moreover, as (Hω)|�cL

is the Dirichlet restrictionof HP

ω,L, its eigenvalues are larger than those of HPω,L. In particular, its

second eigenvalue is larger than E1(ω). Hence, up to a small loss inprobability, we may assume it is larger than E0(ω) + L−ν as we know theestimate (24). This implies that we may assume that E0(ω) is the groundstate of (Hω)�c

j.

So we obtain{

ω; EP1 [ω, L] − EP

0 [ω, L] � η L−d,

|x0(ω) − x1(ω)| � λ log L

}

⊂ 1 ∪⋃

1� j�J

j

where 1 = \ (I,δ,L ∪ {ω; E1[ω, L] − E0[ω, L] � L−ν} and

j ={ω; dist(σ ((Hω)|C j+�λ log L/4), inf σ((Hω)|�c

j)) � ηL−d + L−ν

}

As (Hω)|C j+�λ log L/4 and (Hω)|�cj

are independent of each other, we estimatethe probability of j using Wegner’s estimate to obtain

P[ j] � C(ηL−d + L−ν)(log L)d.

Hence, one obtains

P

[E1[ω, L] − E0[ω, L] � η L−d,

|x0(ω) − x1(ω)| � λ log L

]

� C(ηL−d + L−ν)(log L)d Ld

(log L)d+ 2L−p

� C(η + L−p)

if ν > d + p.This completes the proof in the discrete setting. To prove Lemma 13 for the

continuum case, one does the same modifications as in the proof of Lemma 12in the continuum setting.

Setting ε = d(1/ξ − 1), Proposition 10 then follows from Lemmas 12 and 13.


4 Proof of Theorem 3

Defining π0 = |ϕ0〉〈ϕ0| and applying the definition of the ground state, we canestimate

EP1

[ω, L

] ∥∥(1 − π0) ϕGP

∥∥ + EP

0

[ω, L

] ∥∥π0ϕ

GP∥∥

� EGPω,L

∥∥(1 − π0)ϕ

GP∥∥ + EGP

ω,L

∥∥π0ϕ

GP∥∥ ,

respectively(EP

1

[ω, L

] − EGPω,L

) ∥∥(1 − π0)ϕ

GP∥∥ �

(EGP

ω,L − EP0 [ω, L]) ∥∥π0ϕ

GP∥∥ .

As a consequence of Propositions 4 and 10, we know with a probabilitylarger than 1 − (Cη + L−p) that, for η ∈ (0, 1) and fd defined in (3) theestimates

EP1 [ω, L] − EGP

ω,L � EP1 [ω, L] − EP

0 [ω, L] � ηL−d [1 + (log L)d−2/d+ε]−1

and

EGPω,L − EP

0 [ω, L] � CU fd(log L)

are satisfied. We obtain∥∥(1 − π0)ϕ

GP∥∥ � CU fd(log L)η−1Ld [1 + (log L)d−2/d+ε

] ∥∥π0ϕ

GP∥∥

and∣∣〈ϕ0, ϕ

GP〉∣∣2 = ∥∥π0ϕ

GP∥∥2 = 1 − ∥

∥(1 − π0)ϕGP∥∥2

� 1 − [CU fd(log L)η−1Ld [1 + (log L)d−2/d+ε

]]2.

Applying the assumption concerning the coupling constant U i.e.

U = U(L) = o(

L−d [1 + (log L)d−2/d+ε]−1 [

fd(log L)]−1

)

and setting

η = η(L) =√∣∣U(L)Ld

[1 + (log L)d−2/d+ε

]fd(log L)

∣∣

we get that, η(L) → 0 when L → +∞ and for some C > 0,

P({

ω; ||〈ϕ0, ϕGP〉| − 1| � Cη(L)

})� C

(η(L) + L−p) .

This completes the proof of Theorem 3.

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